Explicit Filtering and Reconstruction Turbulence Modeling...

Explicit Filtering and Reconstruction Turbulence Modeling for Large-Eddy Simulationof Neutral Boundary Layer Flow

FOTINI KATOPODES CHOW AND ROBERT L. STREET

Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University,Stanford, California

MING XUE

School of Meteorology, and Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma

JOEL H. FERZIGER

Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University,Stanford, California

(Manuscript received 15 April 2004, in final form 23 September 2004)

ABSTRACT

Standard turbulence closures for large-eddy simulations of atmospheric flow based on finite-difference orfinite-volume codes use eddy-viscosity models and hence ignore the contribution of the resolved subfilter-scale stresses. These eddy-viscosity closures are unable to produce the expected logarithmic region near thesurface in neutral boundary layer flows. Here, explicit filtering and reconstruction are used to improve therepresentation of the resolvable subfilter-scale (RSFS) stresses, and a dynamic eddy-viscosity model is usedfor the subgrid-scale (SGS) stresses. Combining reconstruction and eddy-viscosity models yields a sophis-ticated (and higher order) version of the well-known mixed model of Bardina et al.; the explicit filtering andreconstruction procedures clearly delineate the contribution of the RSFS and SGS motions. A near-wallstress model is implemented to supplement the turbulence models and account for the stress induced byfiltering near a solid boundary as well as the effect of the large grid aspect ratio. Results for neutralboundary layer flow over a rough wall using the combined dynamic reconstruction model and the near-wallstress model show excellent agreement with similarity theory logarithmic velocity profiles, a significantimprovement over standard eddy-viscosity closures. Stress profiles also exhibit the expected pattern withincreased reconstruction level.

1. Introduction

In large-eddy simulation (LES), the fluid field is spa-tially filtered to separate large eddies from smaller mo-tions; the larger scales are simulated accurately, whilethe effect of the smaller, subfilter scales on the largescales is modeled. The construction of an appropriatesubfilter-scale (SFS) turbulence model for LES hasbeen the subject of decades of research. In this paper,we adopt a velocity partitioning approach (Carati et al.2001), which facilitates the proper modeling of the SFSstresses when finite-difference or finite-volume meth-ods are used. This approach has proven advantageousin low Reynolds number flows (Gullbrand and Chow

2003). We extend the velocity partitioning idea to large-eddy simulation of the atmospheric boundary layer,where the situation is complicated by high Reynoldsnumber flow and a rough lower boundary. The effect ofthe latter must be dealt with by use of a wall model.This paper focuses on neutrally stratified flow over flatterrain, but the method can be used for general flowsituations.

The definition and application of the LES filters dif-ferentiate the velocity partitioning approach from tra-ditional methods. The traditional LES approach treatsthe grid as the filter that separates large and small mo-tions. The nature of this filter is both unknown anddifferent for each term in the equations. The effect ofthe small motions on the larger scales is modeled as onepiece, usually with an eddy-viscosity model. In contrast,the velocity partitioning approach applies a smooth ex-plicit filter (such as a tophat or a Gaussian filter) toseparate resolved and SFS motions. It furthermore rec-

Corresponding author address: Fotini Katopodes Chow, Atmo-spheric Science Division, Lawrence Livermore National Labora-tory, P.O. Box 808, L-103, Livermore, CA 94551.E-mail: [email protected]

2058 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62

© 2005 American Meteorological Society

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ognizes that the presence of the numerical grid dividesthe subfilter-scale motions into resolved and unre-solved portions. The resolvable subfilter-scale (RSFS)motions can be reconstructed using series expansions,while the unresolvable subfilter-scale (USFS) motions[also called subgrid scale (SGS)] are modeled sepa-rately, typically with an eddy-viscosity model. Thismethod is also known as “explicit filtering and recon-struction” (Gullbrand and Chow 2003). Both the RSFSand SGS models rely on knowledge of the resolved-scale behavior alone, but the RSFS portion can be ob-tained directly by an inverse filtering operation, as ex-plained in detail later.

Reconstruction of the RSFS stresses requires thedefinition and application of an explicit filter in theLES computation. The filter must be smooth (in wavespace) and have a width larger than the grid spacing.The explicit filter is used in the series expansions in thereconstruction procedure and also serves to damp fi-nite-difference and aliasing errors (Lund 1997; Chowand Moin 2003). Thus, explicit filtering and reconstruc-tion are especially useful for finite-volume or finite-difference codes, which are easily applied to flows overcomplex geometries of interest in atmospheric bound-ary layer studies. For de-aliased spectral methods, ex-plicit filtering and reconstruction provide no advantageover the traditional approach (see Winckelmans andJeanmart 2001) because numerical errors are easilycontrolled.

A well-known problem in atmospheric boundarylayer simulations is their lack of agreement with simi-larity theory in the near-wall region. In this region ofthe flow, the grid resolution is not adequate to resolveenergy-containing scales and the contribution of theSFS model dominates that of the resolved terms [seethe discussions in Sullivan et al. (1994); Khanna andBrasseur (1997); Kosović (1997); Juneja and Brasseur(1999)]. Standard turbulence closures for atmosphericboundary layer flows use eddy-viscosity models alone(ignoring the contribution of the resolvable subfilter-scale stresses) and do not have correct near-wall behav-ior. In this paper we explore whether explicit filteringand reconstruction can improve LES results, as dem-onstrated in low Reynolds number flows (Gullbrandand Chow 2003). The rough bottom boundary in theatmospheric boundary layer requires special treatment(e.g., specifying a log law approximate boundary con-dition); this will be addressed in the context of velocitypartitioning.

We examine a specific test case: the neutral, rotation-influenced, large-scale boundary layer flow consideredby Andren et al. (1994). A state-of-the-art atmosphericmesoscale and small-scale simulation model is used,namely, the Advanced Regional Prediction System(ARPS; Xue et al. 1995, 2000, 2001). ARPS is a finite-difference LES-capable code designed for flow over ir-regular terrain, so spectral methods and sharp Fouriercutoffs in filters are not viable options. The enhance-

ments to the code are those associated with the newsubfilter-scale models [see also Chow and Street (2002)and Chow (2004)].

Our goal is to learn what logical steps and proceduresare required for subfilter-scale modeling to bring thesimulated flow fields into agreement with theoreticalexpectations in the near-wall region. This includes de-termining appropriate turbulence models and treat-ments for the rough lower boundary. As a test case forthe traditional LES approach, we use the Smagorinskymodel (Smagorinsky 1963), which despite its deficien-cies, remains commonly used in atmospheric applica-tions. For the explicit filtering and reconstruction ap-proach, we use the approximate deconvolution ap-proach of Stolz and Adams (1999) for the RSFS stresseswith the dynamic eddy-viscosity model of Wong andLilly (1994) to represent SGS motions; these are thecomponents of our dynamic reconstruction model(DRM), similar to that used by Gullbrand and Chow(2003). The properties of the reconstruction model arealso discussed by use of the Taylor series expansionmodel of Katopodes et al. (2000a,b). We also compareresults using explicit filtering but ignoring the RSFSterm, using the dynamic eddy-viscosity model (Wongand Lilly 1994) alone. Three-dimensional filters areused so that our approach is general enough for flowover complex terrain where horizontal planar averagesare not applicable. The dynamic eddy-viscosity modelsrequire augmentation near the lower boundary, so thenear-wall shear stress term of Brown et al. (2001) isincluded to account for the stress induced by filteringnear a solid boundary and for the effect of the large gridaspect ratio typically found near the boundary (Du-brulle et al. 2002; Nakayama and Sakio 2004). Thisnear-wall stress term can be considered part of the SGSmodel.

Combining reconstruction and eddy-viscosity modelsyields a generalized (and higher order) version of thewell-known mixed model of Bardina et al. (1983), butnow the explicit filtering and reconstruction proceduresdelineate clearly the contribution of the RSFS and SGSmotions. We advocate the general explicit filtering andreconstruction approach. The specific components ofthe models can vary, and there remains room for muchfurther research, particularly for SGS (including near-wall) models. The RSFS component, as will be shown,can be successfully recovered by series expansion meth-ods. To the authors’ knowledge, explicit filtering andreconstruction approaches have not previously beenapplied to large-scale flows over rough surfaces.

The following sections summarize the framework forconstruction of a hybrid, or mixed, LES SFS closuremodel using separate RSFS and SGS components. Wealso describe the implementation of the models andresults from LES simulations of the neutral boundarylayer. The results using the combined DRM and near-wall stress models show excellent agreement with simi-larity theory logarithmic velocity profiles, which is a

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significant improvement over standard eddy-viscosityclosures.

2. Decomposition of subfilter-scale stresses

To facilitate our understanding of the requirementsin SFS modeling and especially to improve turbulencemodels in the near-wall region, it is useful to considervelocity partitioning schemes such as those of Carati etal. (2001), Zhou et al. (2001), and Hughes et al.(2001a,b). Figure 1 shows a schematic similar to that ofCarati et al. (2001) of a typical energy spectrum from aturbulent flow. The application of a filter (which issmooth in wave space) and a discretization operator(needed to solve the LES equations on a discrete grid)separates the spectrum into three parts. The low wave-numbers are filtered and well resolved on the grid.They are contained in the velocity ũi, where the tildeoperator represents the effect of discretization and theoverbar an explicit smooth filter. The middle portion(shaded) represents subfilter-scale motions that are be-tween the filter and grid cutoffs and hence resolvableon the grid. These resolved subfilter-scale motions cantheoretically be reconstructed by an inverse filter op-eration. Reconstruction is limited, however, by numeri-cal errors (NE), which increase near the grid cutoffbecause of the modified wavenumber effect on finite-difference evaluation of derivatives (Moin 2001). In ad-dition, there are errors due to aliasing effects, thoughthese are not as important in finite difference as inspectral codes (Chow and Moin 2003). [Note that theexplicit filter damps these numerical errors so that rep-resentation of the resolved field is more accurate(Chow and Moin 2003).] The portion to the right of the

vertical dashed line contains subgrid-scale motions thatcannot be resolved on the grid and must be modeled.These are motions that are smaller than the Nyquistwavenumber cutoff, kg � 2�x. The amount of energycontained in the SGS portion depends on grid resolu-tion and the distance from the wall (Khanna and Bras-seur 1997). In our chosen notation, the tilde operatorincludes all effects due to numerics (the Nyquist cutoffand numerical errors); we use the symbol loosely be-cause discretization effects are not exactly known. Wealso emphasize that Fig. 1 is an instantaneous represen-tation of the separation of scales; time evolution willalter the extent to which RSFS motions can be accu-rately reconstructed, as discussed later.

ARPS employs the spatially filtered compressiblenonhydrostatic Navier–Stokes equations. For this pa-per, ARPS was operated in a quasi-incompressiblemode (Xu et al. 1996). Using the notation definedabove, the LES governing equations for the resolvedvelocity are

�ũi�t

��ũiũj�

�xj� �

1�

�p̃

�xi� g�i3 � �imnfnũm �

��̃ij�xj

�1�

�ũi�xj

� 0, �2�

where viscous terms have been neglected. Here ũi arethe velocity components, p̃ the pressure, � the density,and f the Coriolis parameter. While the discretizationeffects are different for every term in the equation (dueto the various finite-difference schemes used), the sameexplicit filter is applied to all variables. It is assumedthat the filtering and discretization operations commutewith the spatial derivatives, which is true for spatiallyhomogeneous filters. Some error is introduced if this isnot so (Ghosal and Moin 1995). Further details on theequations used by ARPS are given in Xue et al. (2000)and Chow (2004, appendices C and D). The configura-tion used for this work is described further in section 5a.

We define the total SFS stress as

�ij � uiuj � ũiũj. �3�

Note that when the SFS stress appears in the filteredand discretized Navier–Stokes equations, it appears as(̃ij /xj), where the tilde indicates the added effect ofthe discretization, as in the advection terms. [The dis-cretization operation is essentially a de-aliasing step aswell (Chow and Moin 2003)]. The full turbulent stresscan be decomposed into resolved and unresolved por-tions:

�SFS � �ij � uiuj � ũiũj � �uiuj � ũiũj��

SGS

� �ũiũj � ũiũj��

RSFS

.

�4�

The first two terms on the right-hand side are the sub-grid-scale stresses, SGS. They depend on scales be-

FIG. 1. Schematic of velocity energy spectrum (with linear axes)showing partitioning into resolved, RSFS, and SGS motions. TheNE region is also shown above the dotted line. The grid is indi-cated by the vertical dashed line at wavenumber kg (correspond-ing to the minimum resolvable wavelength), and the explicit filterby the curved dashed line.


yond the resolution domain of the LES and contain theunclosed nonlinear term uiuj, which must be modeled.The last two terms are the filtered-scale stresses, RSFS,which depend on the differences between the exact andfiltered velocity fields within the resolution domain.This resolved subfilter-scale component, RSFS, cantheoretically be reconstructed because it is a function ofũi, which can be obtained by deconvolution (inversefiltering).

Currently available SGS models do not represent thetrue SGS motions well. It is hoped that by reconstruct-ing the RSFS portion, the overall representation of theSFS stress will be improved, as the SGS contributionwill decrease overall. However, near a rough wall, eddysizes decrease much faster than any grid stretching, andthe bulk of the stress contribution comes from SGSterms (Sullivan et al. 2003). In general, the total stressis given by

�total � �Resolved � �RSFS � �SGS, �5�

where Resolved is the resolved-scale stress defined insection 5c3. Because eddies scale roughly as distancefrom the boundary and the filter and grid cutoffs arefixed, as we approach the wall (z → 0), it must be that

Resolved → 0 and RSFS → 0 because there are no eddiesin their respective spectral areas and all of the eddiesare subgrid, so total → SGS as z → 0. The stress on thewall is given by a wall model such as the log law. Thus

SGS supports the total stress at the wall, suggesting theuse of a specific near-wall stress model that representsthe stress induced by the rough boundary. We employa separate near-wall stress model as a supplement tothe general SGS models used in this work.

3. Reconstruction models: Series expansionapproach

Using this velocity partitioning framework for theturbulence closure, the RSFS and SGS components canbe modeled separately. We first focus on the RSFScomponents, which can be reconstructed in terms of theresolved velocity. Several methods have been proposedto represent such subfilter-scale motions. Bardina et al.(1983) made a seminal contribution by introducingscale-similarity models, which create an approximationto the full velocity field to estimate the RSFS stress.In Bardina’s model, the discrete full velocity is ap-proximated by the filtered velocity, ũi � ũi, to obtain

RSFS � ũiũj � ũi ũj.

Later models have included those of Yeo and Bed-ford (1988), Shah and Ferziger (1995), Geurts (1997),Stolz and Adams (1999), Zhou et al. (2001), and Du-brulle et al. (2002) [see the review of Domaradzki andAdams (2002)]. All of these methods, except the lasttwo, rely on approximate filter inversion. The methodof Zhou et al. (2001) models motions beyond a sharpfilter cutoff with an evolution equation; in our context,

this approach may be useful as an SGS closure. Du-brulle et al. (2002) also model scales smaller than thegrid using a separate dynamic equation. Here, we focuson velocity reconstruction approaches using Taylor se-ries expansions (Katopodes et al. 2000b) and the vanCittert iterative method used in the approximate de-convolution model of Stolz et al. (2001a). These seriesexpansions rely on the application of a smooth explicitfilter; a spectral cutoff filter cannot be used.

a. Recursive Taylor series expansions

Katopodes et al. (2000a,b) used successive inversionof a Taylor series expansion to express the unfiltered(but resolvable) velocity in terms of the filtered veloc-ity. For an isotropic tophat filter applied on a uniformgrid, the expansion reduces to

ũi�x, y, z� � ũi�x, y, z� ��f

2

24�2 ũi

�7�f

4

5760��4 ũi�x4 � �4 ũi

�y4�

�4 ũi�z4

��

5�f4

1728� �4 ũi�x2�y2 � �4 ũi

�y2�z2�

�4 ũi�x2�z2

�� O��f

6�, �6�

to fourth order in the explicit filter width, �f. An aniso-tropic filter is used in simulations, but the isotropic ver-sion is shown here for simplicity. Odd powers of x, y, orz disappear because of symmetry. Other spatially com-pact filters give similar results, with a change in theexpansion coefficients. The expansion can be extendedto an arbitrary order of accuracy by including moreterms in the series, though these become cumbersometo compute. Note that the series expansion only recov-ers scales up to the grid cutoff. The approach can simi-larly be applied to the scalar transport equation, asdone by Katopodes et al. (2000a).

b. Approximate deconvolution method

High-order reconstruction of the RSFS stress tensorcan also be achieved with the iterative deconvolutionmethod of van Cittert (1931). This reconstruction isused by Stolz and Adams (1999) and Stolz et al. (2001a)who call their RSFS model the approximate deconvo-lution model (ADM). The unfiltered quantities can bederived by a series of successive filtering operations (G)applied to the filtered quantities with

ũi � ũi � �I � G� * ũi � �I � G� * ��I � G� * ũi � . . . ,

�7�

where I is the identity operator, and G is the explicitfilter. This expansion can also be extended to an arbi-trary order of accuracy by including more terms in theseries, though it is not immediately obvious what the

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order of magnitude of the next set of terms is. Level-nreconstruction includes the first n � 1 terms of theseries. Computation of higher-order terms is straight-forward, as it simply requires repeated application ofthe same filter operator.

c. Generation of the RSFS models

We derive models for ij by substituting a series ex-pansion for the reconstructed velocity (ũ*i ) directly into(4) to obtain

�RSFS � ũ*i ũ*j � ũ*i ũ*j . �8�

The series expansions for ũi could also be substituteddirectly into the filtered advection terms of the filteredNavier–Stokes equations, without moving the filteredterms to the right-hand side. This was done by Gull-brand and Chow (2003) and Stolz et al. (2001a), whilethe SGS contribution was added to the right-hand sideas usual. In our implementation, we have both the se-ries expansions and the SGS models on the right-handside for ease of implementation in ARPS.

For the ADM approach, nothing further is required.In the Taylor series approach, we expand Eq. (8) toderive RSFS models of arbitrary order of accuracy inthe (isotropic) filter width, �f, giving (to fourth order):

�RSFS � ũiũj � ũiũj ��f

2

24ũi�

2 ũj ��f

2

24ũj�

2 ũi

��f

2

24ũi�

2 ũj ��f

2

24ũj�

2 ũi. �9�

The first two terms are analogous to the Leonard termsin the SFS stress; the higher-order derivative terms canbe shown to be dissipative (Clark et al. 1977). To sec-ond order in the filter width, Eq. (9) reduces to theBardina scale similarity model, as does the ADM atlowest order.

The series model in Eq. (9) can be rearranged byexpanding the higher derivatives and using the filterdefinition to “unfilter” a few terms. We obtain

�RSFS ��f

2

12�ũi�xm

�ũj�xm

, �10�

which is equivalent to Eq. (9) to fourth order in thefilter width [see Katopodes et al. (2000b); Chow (2004)for details]. Equation (10) is similar to the model pro-posed by Clark et al. (1977) [also known as the tensor-diffusivity model, see, e.g., Winckelmans et al. (2001)]but with an extra filter. This modified Clark model isconsiderably simpler than Eq. (9) to implement nu-merically. We have not, however, presented results forthis model here because it can be shown to be equiva-lent to the ADM (see below) and does not performbetter than the ADM, which is easier to implement.The Taylor series method is, however, useful for relat-ing the truncation error to the filter width, which leads

to the important property that the series models satisfythe ij evolution equations (see below). [For results us-ing Eq. (10), see Chow (2004); see Winckelmans et al.(2001) and Iliescu and Fischer (2003) for other imple-mentations of the tensor-diffusivity model.]

d. Properties of the reconstructed RSFSA few observations about both reconstruction meth-

ods [Eqs. (6) and (7)] are worthwhile. First, neithermodel has parameters other than the filter width andthe number of terms to keep in each series, and noassumptions are made about the form of the RSFS mo-tions. The models should thus be able to capture aniso-tropic motions. Scale-similar models are also invariantunder Galilean transformations (Speziale 1985), andthey exhibit correct near-wall behavior in smooth-wallflows when used in a mixed model (Sarghini et al.1999). Their scale-similarity properties have desirableeffects in a priori tests (Katopodes et al. 2000b; Stolz etal. 1999). Furthermore, the evolution equation devel-oped by Katopodes et al. (2000b) for the approximate

ij indicates that these resolvable subfilter-scale stressesare influenced by buoyancy, Coriolis, diffusion, pres-sure, and advection terms, just as the resolved velocitiesare. Thus the expressions in Eqs. (9) and (10) for ij(and as noted below, the equivalent ADM expressions)capture the effects of all of the relevant physical mecha-nisms, to fourth order in the filter width.

While it is not as easy to relate the truncation error ofthe ADM approach to the filter width, the van Cittertiterative method can easily be related to the Taylorseries reconstruction method. For example, appendix Bof Stolz et al. (2001a) shows how the tensor-diffusivitymodel can be derived from the ADM. Using finite-difference representations, it is also easy to show theequivalence of these two reconstruction approaches(Chow 2004). A difference in implementation ariseswhen the reconstructed velocities are substituted backinto the RSFS stress expressions. In the Taylor seriesapproach, terms of fourth-order or higher are explicitlydisregarded, for example, in the computation of theproduct terms. That is, we are careful to maintain theorder of the final expression even after it has been sub-stituted into ij. Discretization is applied only after thefinal form has been obtained. When the ADM recon-struction terms are used, the discretely reconstructedvelocities are substituted directly into the RSFS expres-sion. Therefore, higher-order terms are implicitly con-tained in the RSFS expressions. The difference in theADM (n � 1) versus Taylor series (fourth order) ap-proaches is in the fourth order truncation terms. Theuse of the Taylor series expansions more easily pre-serves the desired order of the reconstruction, but theADM approach is much simpler to implement numeri-cally.

In summary, the series expansion representations ofthe RSFS stresses are theoretically excellent [furtherdiscussion can be found in Gullbrand and Chow (2003)


and Chow (2004)]. They perform well in a priori testsand can be used up to any order of accuracy desired,within the grid resolution limitations. The computa-tional cost does, however, increase with increasing lev-els of reconstruction. Furthermore, interaction with nu-merical errors in a posteriori tests is not easy to predict;results are examined in the later sections. In the nextsection, we describe approaches to modeling SGS mo-tions, which become increasingly important at coarsegrid resolutions such as those present near the lowersurface in atmospheric boundary layer flow simula-tions.

4. SGS and wall models

Aside from the discretization errors in the numericalmodel and in the reconstruction procedure itself, theabove reconstruction methods are exact to within thechosen truncation error. Unfortunately, the turbulenceclosure problem remains in the SGS terms. Equation(4) still has the unclosed term uiuj in the SGS portion ofthe total SFS stress. The interaction of the SGS andRSFS motions is also hard to predict and may createerrors that limit the reconstruction as well. The SGSterm must account for any such errors in the RSFSterm.

Carati et al. (2001) suggest that there is a separationof scales of sorts between the resolved component andthe subgrid component of the velocity field (see Fig. 1)that perhaps makes an eddy-viscosity model a goodchoice for the SGS motions. Unfortunately, they findvery poor correlations between eddy-viscosity modelquantities and those calculated from direct numericalsimulation (DNS) data. For lack of a better framework,a simple eddy-viscosity form is also assumed in thiswork for modeling the unclosed SGS term:

�SGS � �2�TS̃ij, �11�

where �T

is the eddy viscosity, and S̃ij � (1/2)(ũi /xj �ũj /xi) is the resolved strain rate tensor. The closureproblem shifts to determining the best representationfor �

T. Despite the known shortcomings of this class of

models, they are convenient to use when energy trans-fer to the subgrid scales is desired. The variations ofeddy-viscosity models are too numerous to describethem all, so we focus on the models we have chosen forour study. These particular SGS models have been usedbefore to represent the complete SFS stress, but nonehave been previously used with explicit filtering andreconstruction in large-scale flows. Near the lowerboundary, the SGS closure requires special treatment,as described in section 4d below.

a. Smagorinsky-based eddy-viscosity models

One of the most commonly used eddy-viscosity mod-els is the Smagorinsky model (Smagorinsky 1963),which assumes

�T � �CS�g�2�2S̃ij S̃ij�

12, �12�

where CS is the Smagorinsky coefficient, and �g is thegrid spacing. The Smagorinsky model has several draw-backs, especially near the surface, where it overpredictsthe stresses (see the discussion in Juneja and Brasseur1999). All eddy-viscosity models fail to allow for back-scatter of energy from small to large scales. Atmo-spheric measurements show that backscatter is presentnear the surface and should be included in LES turbu-lence closure schemes [see Porté-Agel et al. (2001) andSullivan et al. (2003)]. Correlations from a priori testsshow that the eddy-viscosity stress tensors are notaligned with calculations from DNS data (Katopodes etal. 2000b). Furthermore, Mason and Thomson (1992)showed that the failure of the Smagorinsky model nearthe surface could not be cured by increasing grid reso-lution. There will always be a region near the roughbottom boundary where the flow is underresolved. Sev-eral modifications and alternatives to improve perfor-mance near the surface have been proposed [see e.g.,Sullivan et al. (1994)], but we use the Smagorinskymodel in its standard form to simplify the comparisons.

b. Dynamic eddy-viscosity models

The standard Smagorinsky model requires the use ofa predetermined coefficient, CS. An alternative is thedynamic Smagorinsky model (DSM), where the Sma-gorinsky coefficient CS is determined automatically byapplying the same models at the test filter level (Ger-mano et al. 1991). The assumption is that the sameSmagorinsky coefficient could be used if the sameequations were applied on a coarser grid. The DSM iswidely used as the eddy-viscosity SGS model in small-scale turbulence studies (see e.g., Germano et al. 1991;Meneveau and Katz 2000). The DSM has been usedwith explicit filtering in small-scale flows by Winckel-mans et al. (2001) and Gullbrand and Chow (2003); thisrequires special attention to the application of the testfilters.

Applications of dynamic models to high Reynoldsnumber flows have been few. Balaras et al. (1995) usedthe DSM in simulations of high Reynolds number chan-nel flow with rough walls (though still at laboratoryscales). Esau (2004) simulated the large-scale neutralatmospheric boundary layer using the DSM and thedynamic mixed model (DMM) of Zang et al. (1993)with some improvement over the standard Smagorin-sky model. Porté-Agel et al. (2000) developed a scale-dependent dynamic Smagorinsky model to account forthe near-wall region where motions are underresolved.When applied to a neutral atmospheric boundary layer,they achieved improved logarithmic velocity profiles,but at the cost of an extra filtering step in the dynamicprocedure and an empirical function to determine thescale dependence.

Because of sensitivities of the DSM to the bottom

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boundary condition (see section 5b), as well as for easeof implementation, we instead use the dynamic modelof Wong and Lilly (1994), who simulated the convectiveboundary layer. Wong and Lilly (1994) present a sim-plified base model that requires no calculations of S̃ijduring the dynamic procedure. Instead, the base modelis derived from Kolmogorov scaling, which expressesthe eddy viscosity as

�T � C23�f

43�13, or �T � C��f43, �13�

where C� � C2/3�1/3 is the coefficient of interest; hence

the traditional requirement that the dissipation rate �be equal to the SGS energy production rate (includingthe buoyancy term) is avoided. This base model re-duces computational cost because the derivatives in Eq.(12) (at grid and test levels) are no longer required. Inthe dynamic Wong–Lilly (DWL) formulation, the coef-ficient is determined using the least squares method ofLilly (1992):

2C��f43 �

�Lij S̃ijc��

�1 � ��t �f�43�S̃ij

c�S̃ij

c��

. �14�

Here, the brackets � � denote local averaging, ∧ denotesthe test filter, and

Lij � ũiũj��c

� ũic� ũjc��c �15�

S̃ijc�

�12��ũic�

�xj�

�ũjc�

�xi� . �16�

We use the notation of Carati et al. (2001) to define thetest-filtered terms; the �c operator denotes the effect ofthe discretization operator at the coarser test-level grid.The discrete forms of the explicit and test-filter opera-tors are given in Eqs. (23)–(25).

c. Combined RSFS and SGS models

We use the ADM (for the RSFS) and the DWL (forthe SGS) to obtain the total SFS stress:

�ij � ũ*i ũ*j � ũ*i ũ*j � 2C��f43S̃ij, �17�

which we call the dynamic reconstruction model (simi-lar to Gullbrand and Chow 2003). We also use ADMreconstruction series from levels zero through ten, de-noted DRM-ADM0–DRM-ADM10. Note that DRM-ADM0 is similar to the DMM with explicit filtering asimplemented by Vreman et al. (1994), but with theDWL substituted for the DSM. The standard Smagor-insky model without an RSFS component is our refer-ence case.

The eddy-viscosity coefficient in this combinedRSFS/SGS approach is determined dynamically from

2C��f43 �

��Lij � Hij�S̃ijc��

�1 � ��t �f�43�S̃ij

c�� S̃ij

c�, �18�

where we define (shown here for level-zero reconstruc-tion only)

� � ��Hij � �ũi

c� ũjc��

c

� ũic�

�ũi

c��c

� � �ũi ũj�c

� ũiũj�c

�, �19�and ask for the reader’s indulgence for the use of suchcompact notation. This derivation again uses the leastsquares approach and is similar to that of Zang et al.(1993) and Vreman et al. (1994), thus taking into ac-count the contribution of the scale-similarity portion (inHij) while computing the dynamic coefficient. The gridcutoff (�, �c) operators are not explicitly applied be-cause no robust means exist for applying a cutoff filterin a finite-difference simulation like ours. The notationserves as a reminder of the effect of the discretizationoperators. The finite-difference schemes on the coarsergrid level limit aliasing effects because of the modifiedwavenumbers that decay to zero near the grid cutoff.Our combined RSFS and SGS models constitute amixed model that represents both backscatter of small-scale energy to the larger scales and forward scatter(dissipation) of large-scale energy by the small scales.Both processes are essential for reasonable representa-tion of subfilter-scale effects.

Finally, we note that the mixed model (RSFS/SGS)approach could be used with other SGS models, such asstatic Smagorinsky, the modified version of Sullivan etal. (1994), or 1.5-order turbulent kinetic energy–based(Deardorff 1980; Moeng 1984) closures commonly usedin cloud-scale simulations with parameterizations ofstability effects. The values of the turbulent kinetic en-ergy (TKE) equation coefficients are often debated(see e.g., Takemi and Rotunno 2003; Deardorff 1971),so a dynamic procedure like that of Wong and Lilly(1994) is more desirable. Stolz et al. (2001a) use theADM reconstruction with a relaxation term to drainenergy at the smallest scales and also include a dynamicprocedure for determining the coefficient. This combi-nation has performed well in simulations of small-scaleincompressible channel flows (Stolz et al. 2001a), andcompressible flows such as shock–turbulent-boundarylayer interaction (Stolz et al. 2001b). For this work, wechose familiar eddy-viscosity formulations for theirknown performance in large-scale flow simulations.

d. Enhanced near-wall stress model

Very near the rough lower boundary, the physics ofthe flow, the grid aspect ratio, and the effect of filtering(in three dimensions) lead to the requirement for spe-cial treatment in the turbulence model. In this near-wallregion, the eddy size decreases much more rapidly thanthe grid spacing (Zhou et al. 2001; Sullivan et al. 2003).The vertical grid spacing is invariably smaller than the


horizontal one. Scotti et al. (1997) examined the dy-namic Smagorinsky model in the context of high-aspectratio grids where two directions are poorly resolvedcompared to a third (x, y compared to z in our case).They suggest that for such grids, higher-wavenumbermodes do not have access to all of the local triadicinteractions and that this affects the energy spectrum.

Thus, while eddies may be relatively well resolved inthe vertical, they are not so in the horizontal, henceintroducing errors. Because 2�x is the minimum eddysize resolved in the horizontal (arguably 4�x is thesmallest well-resolved eddy), over a vertical distance of2�x, eddies of this size are still underresolved. This lackof resolution implies that an additional stress term maybe needed near the wall to represent these motionsbecause the chosen SGS models (described above) areinadequate in this very near surface region.

Furthermore, the filtering process and the physicalexistence of subgrid roughness alter the distribution ofstresses near the surface. Dubrulle et al. [2002, see theirEq. (16)] show that the filtering process near a smoothsolid boundary generates extra stress terms in the mo-mentum equations near the wall. Nakayama and Sakio(2002) also investigated this filtering process by exam-ining the effects of subgrid roughness with DNS of flowover a wavy bottom boundary consisting of small andlarge wavelengths. Filtering the DNS flow field and thewavy DNS boundary produced a snapshot of an idealLES solution over the large wavelength boundary withsubgrid roughness. The filtered velocities at the surfacein the LES domain were then apparently only influ-enced by the larger wavelength topography, but thesubfilter-scale roughness elements (the smaller wave-lengths in the original DNS boundary) had generatedextra stress near the new, smoother boundary. The sur-face stress, originally distributed over the DNS bound-ary, became distributed over a finite vertical layerabove the smoother LES boundary. Nakayama andSakio (2004) and Nakayama et al. (2004) used formalexplicit filtering in the neighborhood of the rough wallboundary to derive the extra stresslike terms that ap-pear in the equations. Use of DNS data allowed them toexamine the distributions of these terms and to suggestuseful ideas to formulate the near-boundary conditions.

The issues of near-wall resolution and physics and ofthe near-wall filtering point to the need for a near-wallstress model that distributes stresses generated at therough wall over a region near the wall. Somewhat simi-lar behavior arises in flow through and over vegetationcanopies (see e.g., Shaw and Schumann 1992; Patton etal. 2001). The solution in these cases is a canopy modelto distribute fluxes over the plant canopy height to ac-count for the increased drag from the vegetation.Brown et al. (2001) extended the idea of plant canopymodels to flow over rough surfaces, to distributestresses generated at the rough surface over a finitedistance from the wall.

In accordance with the effects described above, our

near-wall stress term is limited to the near-wall region,so we model it separately from the main SGS closure.We have implemented the model of Brown et al.(2001), which can be expressed as a forcing term in thehorizontal momentum equations, �Cca(z) | ũ | ũi, wherei � 1, 2. Here, Cc is a scaling factor and the functiona(z) allows for a smooth decay of the forcing functionas the specified cutoff height, hc, is approached; a(z) isset equal to cos2(�z/2hc) for z � hc and is zero other-wise. When implemented numerically, the enhancedstress is included in the turbulence closure stress term,and therefore is integrated numerically using the trap-ezoidal rule:

�i,near-wall � �� Cca�z� | ũ | ũi dz, �20�where the integration constants are chosen so that

i,near-wall � 0 at the top of the enhanced stress layer.This stress is then directly added to the i3 terms fromthe other model components. The recent work of Na-kayama et al. (2004) and the successful simulation ofthe stable boundary layer over flat, rough terrain byCederwall (2001) suggest that this form is both physi-cally and mathematically reasonable.

5. Large-eddy simulations of neutral boundarylayer flow

To test the performance of the RSFS/SGS partition-ing approach, we use the ARPS code to simulate arotation-influenced neutral boundary layer flow casesimilar to that of Andren et al. (1994). For the laminarcase, or with a constant eddy viscosity, this flow has theEkman spiral as an analytical solution (see Stull 1988,pp. 210–212). The fully turbulent solution has been ex-amined by several researchers, for example, Sullivan etal. (1994), Andren et al. (1994), Kosović (1997), Cole-man (1999; DNS), Porté-Agel et al. (2000; no Coriolisforcing), Ding et al. (2001), Redelsperger et al. (2001),Carlotti (2002), and Esau (2004). Using scale analysis,Blackadar and Tennekes (1968) showed that the near-wall region of the turbulent Ekman layer should followa logarithmic law. Traditional eddy-viscosity closuresdo not give a good logarithmic region near the wall,usually overpredicting the shear in the model with thevelocity too low at the wall, and too high further awayfrom it (Andren et al. 1994).

The following sections compare the explicit filteringand reconstruction approach (RSFS plus SGS) with re-sults from the standard Smagorinsky model available inARPS. The TKE 1.5 closure in ARPS gives results simi-lar to the Smagorinsky model, so they are not shown.The simulation setup and the effects of dynamic recon-struction on the flow solution are described below.

a. Model details and flow setup

ARPS was developed at the Center for Analysis andPrediction of Storms at the University of Oklahoma.

JULY 2005 C H O W E T A L . 2065

Intended mainly for mesoscale and small-scale atmo-spheric simulations, ARPS solves the three-dimen-sional, compressible, nonhydrostatic, filtered Navier–Stokes equations. ARPS is employed here in LESmode by using appropriate turbulence closure models.ARPS details can be found in Xue et al. (1995, 2000,2001). As noted in section 2, the equations have beenslightly modified (as detailed in Xu et al. 1996) to makethem closer to the incompressible case studied by An-dren et al. (1994). After this work was completed, weperformed tests with the original compressible ARPScode; the results were essentially the same as those forthe quasi-incompressible case described here.

The flow is driven by a constant pressure gradientcorresponding to a geostrophic wind of (Ug, Vg) � (10,0) m s�1. ARPS is run for thirty nondimensional timeperiods tf, where f is the Coriolis parameter, set equalto 1 � 10�4 s�1. This configuration results in an Ekman-like spiral for the mean velocities. The initial conditionsare the analytical Ekman spiral solution with small per-turbations that trigger instabilities so that the flow be-comes fully turbulent (see Fig. 2 later).

The grid size for the control case is (43, 43, 43) withgrid spacings of 32 m in the horizontal. In ARPS thiscorresponds to a domain size of �x(nx � 3) � 1280 min each horizontal direction. In the vertical, a stretchedgrid is used, with �zmin � 10 m spacing near the bottomand up to 65 m near the top of the domain. The averagespacing is 37.5 m and the domain height (H) is 1500 m.Fourth-order spatial differencing was used for the ad-vection terms. Temporal discretization uses a mode-splitting technique to accommodate high-frequencyacoustic waves; the large time steps use the leapfrog

method, while first-order forward–backward explicittime stepping is used for the small time steps (exceptfor terms responsible for vertical acoustic propagation,which are treated implicitly). Higher and lower gridresolutions as well as different grid aspect ratios werealso studied. Run parameters are given in Table 1 for allthe tests and the base case.

The top and bottom boundaries are treated as rigidfree-slip (also called semi-slip at the lower boundary).Surface momentum fluxes (e.g., uw | s) are parameter-ized to account for the influence of the rough bottomsurface with an instantaneous logarithmic drag law(used here with constant drag coefficients) at each gridpoint:

uw | s � CDũ�ũ2 � ̃2, �21�

where ũ and �̃ are evaluated at the first grid point abovethe wall (at height �zmin/2). The drag coefficient CD isdetermined by

CD � �� ln�z � z0z0 ��2

, �22�

where � is the von Kármán constant (0.4) and z0 is thebottom roughness height, set to 0.1 m, and again z is setto �zmin/2. For the high-resolution and low-aspect ratiogrids used here, it may be useful to use fluctuating sur-face-flux coefficients (as done by Wyngaard et al. 1998),but they found little improvement from this step be-cause of deficiencies in SGS models in the surface layer.At the lateral boundaries, periodic conditions are usedfor this idealized flat-terrain study.

TABLE 1. List of simulations and parameters. Standard grid sizes are followed by those with various grid aspect ratios.

Run name Grid size �x �zmin Large �t Small �t Cc hc

Smagorinsky* (43, 43, 43) 32 m 10 m 0.5 s 0.05 s — —DWL (43, 43, 43) 32 m 10 m 0.5 s 0.05 s 0.5 4�xDRM-ADM0 (43, 43, 43) 32 m 10 m 0.5 s 0.05 s 0.5 4�xDRM-ADM1 (43, 43, 43) 32 m 10 m 0.5 s 0.05 s 0.5 4�xDRM-ADM2 (43, 43, 43) 32 m 10 m 0.5 s 0.05 s 0.5 4�xDRM-ADM5 (43, 43, 43) 32 m 10 m 0.5 s 0.05 s 0.5 4�xDRM-ADM10 (43, 43, 43) 32 m 10 m 0.5 s 0.05 s 0.5 4�xSmagorinsky (43, 43, 43) 64 m 10 m 1.0 s 0.1 s — —Smagorinsky (43, 43, 43) 128 m 10 m 2.0 s 0.2 s — —Smagorinsky (43, 43, 83) 32 m 5 m 0.5 s 0.05 s — —Smagorinsky (83, 83, 83) 16 m 5 m 0.25 s 0.025 s — —Smagorinsky (83, 83, 83) 32 m 10 m 0.5 s 0.05 s — —DRM-ADM0 (43, 43, 23) 32 m 20 m 0.5 s 0.05 s 0.4 4�xDRM-ADM0 (43, 43, 23) 64 m 20 m 1.0 s 0.1 0.6 4�xDRM-ADM0 (43, 43, 23) 128 m 20 m 2.0 s 0.2 s 0.8 4�xDRM-ADM0 (43, 43, 43) 16 m 16 m 0.25 s 0.025 s 0.4 4�xDRM-ADM0 (43, 43, 43) 64 m 10 m 1.0 s 0.1 s 0.75 4�xDRM-ADM0 (43, 43, 43) 96 m 10 m 1.0 s 0.1 s 0.83 2�xDRM-ADM0 (43, 43, 43) 128 m 10 m 2.0 s 0.2 s 0.85 2�xDRM-ADM0 (43, 43, 83) 32 m 5 m 0.5 s 0.05 s 0.7 2�xDRM-ADM0 (43, 43, 83) 64 m 5 m 0.5 s 0.05 s 0.85 2�xDRM-ADM0 (83, 83, 83) 16 m 5 m 0.25 s 0.025 s 0.5 4�xDRM-ADM0 (83, 83, 83) 32 m 10 m 0.5 s 0.05 s 0.5 4�x

* Base case.


b. Implementation of RSFS, SGS, and wall models

The Smagorinsky closure provides the base standardagainst which we compare our new turbulence model-ing approaches. The Smagorinsky coefficient is chosenas CS � 0.18 (see e.g., Sullivan et al. 1994).

The anisotropic explicit filter for the reconstructionmodels was applied at twice the grid spacing, 2�g; thischoice is consistent with the findings of Ghosal (1996)and Chow and Moin (2003), which dictate that a mini-mum filter-grid ratio of two should be used with fourth-order advection schemes (as in ARPS). This filter ap-pears in the calculation of the RSFS terms and in thedynamic procedure. For ease of implementation, weapply the van Cittert series expansion reconstructionprocedure in computational space, which has a uniformgrid (though still allowing for anisotropy); therefore thefilters are applied in computational space (Jordan 1999;Stolz et al. 2001a). The greatest difficulty in implemen-tation of the RSFS models is near-solid boundaries,where care must be taken in representing higher-orderterms. The values at the wall can be sensitive to thebottom boundary conditions due to the repeated filter-ing that occurs there. We found, however, that whenRSFS models are used together with SGS models at thewall, the RSFS contribution decays to zero (see section2) regardless of the specified boundary conditions.

On the other hand, our tests of the dynamic Smago-rinsky model (DSM) as the SGS model in ARPSshowed very strong sensitivity to the bottom boundaryconditions. Porté-Agel et al. (2000) also performedtests with the standard DSM and obtained values of thedynamic coefficient that were too low near the wall.The dynamic DWL model was not as sensitive to thechosen boundary conditions (because �T does not de-pend on S̃13 explicitly). The stress provided by theDWL model is still too small at the wall, but the result-ing profile is better. In addition to giving more reason-able profiles near the wall, the DWL is easier to imple-ment and more computationally efficient than theDSM. Wong and Lilly (1994) used a horizontal planeaverage, which is not general enough to apply to flowover terrain; we therefore use a local filter created byapplying the test filter twice. The dynamic eddy viscos-ity is also clipped at �1.5 � 10�5 to prevent the eddyviscosity from becoming too negative locally and caus-ing instabilities (see Zang et al. 1993; Wong and Lilly1994); further details are in Chow (2004).

The only input parameter required for the DWL isthe ratio of the test and explicit filters, �t/�f, which ischosen to be two, consistent with the �f/�g ratio chosenabove (Gullbrand and Chow 2003). The resulting testfilter is thus 4�x, which differs from the usual test filterused without explicit filtering. When no reconstructionis used, the test filter is often chosen as twice the gridspacing (see e.g., Zang et al. 1993). The choice of thetest filter width is a distinguishing, though subtle, fea-ture of the explicit filtering approach (Winckelmans et

al. 2001; Gullbrand and Chow 2003). The resulting testplus explicit filter must be similar in shape to the origi-nal explicit filter (Winckelmans et al. 2001). The dis-crete tophat explicit filter is applied (to the streamwisevelocity u, for example) with

ui � 0.25ui�1 � 0.5ui � 0.25ui�1, �23�

where i temporarily denotes the grid index; thereforethe test filter is

ûi � 0.5ui�1 � 0.5ui�1 �24�

to give a combined test plus explicit filter of

ûi � 0.125ui�2 � 0.25ui�1 � 0.25ui � 0.25ui�1

� 0.125ui�2, �25�

which is simply a tophat filter of twice the width of theexplicit filter (for further details see Gullbrand andChow 2003).

Following Brown et al. (2001), Cederwall (2001), andChow and Street (2002), we implement a near-wallstress model, based on the model given previously inEq. (20). Brown et al. (2001) choose a constant Cc sothat the velocity at the top of the canopy matches ex-perimental measurements. Cederwall (2001) selectedCc such that the near-wall stress model augmented thetotal stress at the first grid point above the wall to makeit equal to the total local bottom shear stress. We allowCc to be locally proportional to the bottom shear stressin each horizontal direction. The proportionality factoris chosen so that the near-wall stress model provides thenecessary augmentation that yields logarithmic meanvelocity profiles near the wall. The variation of Cc andhc (the layer height) with grid spacing and aspect ratiois described later. The near-wall stress model is part ofthe SGS model component, but is treated separatelyfrom the eddy-viscosity contribution because it ispresent strictly very near the wall.

Finally, current turbulence models often do not pro-vide enough dissipation at the highest frequencies, socomputational mixing terms may be added for stabilityif desired. We follow the recommendations in ARPSand use a small fourth-order term, which can be con-sidered to be a type of hyper-viscosity. ARPS also in-cludes a divergence damping term to control acousticnoise. The impact of these damping terms has beenassessed and found to be minimal (see spectra plottedin Chow 2004).

c. Simulation results

1) TIME EVOLUTION

Figures 2a and 2b show the evolution of the nonsta-tionarity measures Cu and C� defined by [following An-dren et al. (1994)]

JULY 2005 C H O W E T A L . 2067

Cu � �f

uws�

0

H

�� ̃� � Vg� dz �26�

C �f

ws�

0

H

�� ũ� � Ug� dz, �27�

where H is the stress-free top of the boundary layer andsimulation domain, and uws and �ws denote the totalsurface stresses in the uw and �w planes, respectively.The brackets � � here denote horizontal planar averag-ing using data taken at each time step. At steady state,both Cu and C� should be unity, but even after 100 000s (tf � 10), inertial oscillations are apparent in Figs. 2aand b. These oscillations have a period of 2�/f andresult from the change in balance of pressure gradientand Coriolis forces (Cushman-Roisin 1994). Note thatthe spin-up time is much shorter for the DRM-ADM0run. Andren et al. (1994) state that the inertial oscilla-tions affect primarily the first-order statistics of theflow, but some of our comparisons are quite sensitive,so we seek a solution as close to a statistically steadystate as possible. Unless otherwise indicated, we aver-

age from 200 000 to 300 000 s (tf � 20 to 30) whereoscillations in the C� curve especially have decayed sig-nificantly. Our simulation and averaging times areabout 3 times as long as those of Andren et al. (1994).

2) FIRST-ORDER QUANTITIES

Figure 3a shows semilog plots of mean wind speedfrom four different turbulence model configurations:no model, static Smagorinsky, DWL, and DRM-

ADM0. The wind speed (U � � ũ2 � �̃2) is normalizedby the (time averaged) friction velocity defined by u* �(uw2s � �w

2s)

1/4 (found to be approximately 0.44). Thetheoretical log law profile is also shown. The “nomodel” and Smagorinsky results deviate considerably,but the DWL and DRM results provide good agree-ment. The no model simulation still incorporates therough-wall boundary condition; therefore the surfacestress contributes at the first point above the wall andslows the wind speed.

FIG. 2. Comparison of nonstationarity parameters (a) Cu and(b) C� for the Smagorinsky model and the DRM-ADM0 hybridmodel.

FIG. 3. Comparison of (a) mean wind speed and (b) nondimen-sional mean shear � profiles for no model, the Smagorinskymodel, the dynamic Wong–Lilly model, and for the DRM-ADM0hybrid model (level-0 reconstruction, DWL, and near-wall stress).Theoretical log profile also shown in (a).


A more sensitive measure of a model’s performanceis the nondimensional velocity gradient, �, defined as

� ��z

u*�� ũ�

�z �2 � �� ̃��z �2. �28�Here � is the von Kármán constant, chosen to be 0.4. Ina logarithmic region, � � 1, which we expect for ap-proximately the first 150–250 m above the wall, or10%–15% of the boundary layer depth (0.1–0.15 H)(Sullivan et al. 1994). Profiles of �, shown in Fig. 3b,have been smoothed to remove 2�z waves amplified bythe derivatives (and not visible in the original velocityfields). We see that the overshoot in � reaches 1.6 forthe traditional Smagorinsky model, indicating themodel allows excessive shear near the surface. Similarovershoots were observed in other studies (Andren etal. 1994; Sullivan et al. 1994; Kosović 1997). It appearsthat the Smagorinsky model behaves as a large viscositymodel that mimics the molecular viscosity dissipationnear a smooth wall. In large-scale flows, however, thelinear region is confined to a few millimeters above thesurface, and the log region should cover the entirerange near the wall in our simulation results. Our ef-forts at improving near-surface � profiles are aimed atreplacing this gradient diffusion behavior near the wall.Surprisingly, the results using no model appear fairlylogarithmic after the first point above the wall, but thevelocity magnitude is significantly overpredicted.

When reconstruction and dynamic eddy-viscositymodels are used with the near-wall stress term (DRM-ADM0), values of � within 0.1 of the ideal value(unity) are obtained. This represents a significant im-provement over the standard Smagorinsky simulations.It is better than several previous attempts to improve �profiles, where agreement within about 0.2 from unitywas achieved by Sullivan et al. (1994), the backscatterresults of Mason and Brown in Andren et al. (1994),Porté-Agel et al. (2000), and Ding et al. (2001), usingvarious SGS methods. Our results are comparable tothose of Kosović (1997), where � profiles were within0.1 of unity. The simulations of Esau (2004) using adynamic mixed model (with two-dimensional filtering)did not produce satisfactory results near the wall [seeEsau (2004), their Figs. 11a and 15d]; while the DMMprovides some reconstruction, it was not used with aformal explicit filter, making the level of reconstructionambiguous. Sensitivity to the numerical formulationwith the Smagorinsky model as the base model for thedynamic procedure (as opposed to DWL) and the ab-sence of a near-wall stress model may also have limitedthe improvement obtained by the DMM.

Our hybrid SFS model has the advantage that oncethe filter has been selected, no adjustable parametersare needed aside from those in the enhanced near-wallstress, which is active only in the first few grid pointsabove the wall. The reconstruction component is ofscale-similar form, and thus allows backscatter, that is,

energy flux from the small to the large scales (withinthe grid resolution). This is believed to be especiallyimportant in the atmospheric boundary layer when thelarge scales are not fully resolved and it aids in achiev-ing a logarithmic mean velocity profile. The influenceof the enhanced near-wall stress is felt only near thewall, where it helps to straighten the � profiles. Theimpact of the near-wall stress model does not over-whelm the near-wall results to the point where the tur-bulence model becomes irrelevant. Tests with the samelevel of near-wall stress added to the standard Smago-rinsky model (without reconstruction) did not produceencouraging results (see Fig. 4). Interestingly, the DWLmodel without reconstruction also performs quite wellin these first-order comparisons.

Figure 5 shows the effect of increasing the level ofreconstruction in the DRM-ADM models. The � pro-files indicate that while all the results remain within 0.1of the ideal value, increasing reconstruction moves theprofiles slightly closer to unity very near the wall (z/H� 0.05). Because the curves are so close together, thereis some sensitivity to the time averaging period; Figs.2a,b showed that the deviations from stationarity dif-fered in time with different turbulence models. Wefound that taking a long averaging period (100 000 s)after Cu and C� have decayed significantly gave themost reasonable results. It is not clear, however, that �should be the sole parameter used for comparisons as itis quite sensitive to small wiggles, which are not readilyapparent from the mean velocity profiles in Fig. 3a. Theeffects of adding reconstruction levels become clearerwhen second-order statistics are examined.

3) SECOND-ORDER QUANTITIES

Figure 6 compares the uw stress distribution for theSmagorinsky, DWL, and DRM-ADM0 cases from Fig. 3.

FIG. 4. Comparison of nondimensional mean shear � profilesfor the Smagorinsky model, also using the near-wall stress model,and DRM-ADM0 (with near-wall stress).

JULY 2005 C H O W E T A L . 2069

The resolved uw stress is defined as �(ũ � � ũ �)(w̃ ��w̃ �)�. The Smagorinsky and DWL simulations showsimilar contributions of the SFS stress, which is largestnear the wall. When reconstruction is added, the totalSFS stress increases compared to the eddy-viscositysimulations, while the resolved stress decreases accord-ingly. Similar behavior is observed for the other stresstensor components. The magnitude of the total stress(resolved and SFS) differs slightly between the differ-ent simulations.

The contributions from each SFS stress componentfor DRM-ADM0 are seen in the uw stress profiles inFig. 7. The profiles are also shown on a semi-log plot tomagnify the region near the wall. The total uw stress(resolved plus SFS) is not linear because there is a com-ponent in the �w direction because of Coriolis forcing.

The influence of the near-wall stress model decreaseswith height, becoming zero at the hc � 4�x � 128 m forthis grid. This is equivalent to the minimum well-resolved horizontal eddy size beneath the filter. Weobtain good results with a near-wall stress model pro-portionality factor Cc � 0.5, so that the near-wall stressmodel contributes an amount equal to half the wallstress at the first grid point above the wall. The recon-struction terms (RSFS stress components) decay tozero naturally at the wall in the presence of the othertwo model components (eddy viscosity and near-wallstress; see section 2).

The effects of increasing reconstruction on the uwstresses are shown in Fig. 8. The RSFS component ofthe SFS stress increases with reconstruction level (lev-els 0, 1, and 5), while the SGS component decreasesslightly. Thus, the total SFS contribution increases withincreasing reconstruction (primarily due to the contri-

FIG. 5. Comparison of nondimensional mean shear � profilesfor increasing reconstruction levels: dynamic Wong–Lilly, no re-construction; DRM-ADM0; DRM-ADM1; and DRM-ADM5.

FIG. 6. Comparison of uw resolved (thin lines) and SFS (boldlines) stress profiles for Smagorinsky, DWL, and DRM-ADM0.

FIG. 7. (a) Normalized vertical profiles of the uw total stress forthe DRM-ADM0 model (zero-level reconstruction, dynamicWong–Lilly model, and near-wall canopy stress), with each modelcomponent shown separately in addition to the resolved stress. (b)Same as (a) but with a logarithmic vertical axis to magnify theregion near the wall.


bution of the reconstruction terms), and the resolvedcontribution decreases accordingly (as seen in Fig. 6).Note that the spacing between the level-0 and level-1curves is about the same as that between level-1 andlevel-5, indicating a trend toward convergence. Thistrend was also observed in the small-scale channel flowsimulations of Gullbrand and Chow (2003). The slightlydecreasing contribution of the eddy-viscosity term isalso apparent in Fig. 9, which shows profiles of �T forthe same simulations as in Fig. 5. Near the wall, theeddy viscosity is relatively constant with height. This isthe region where wall effects in the numerical proce-dure are strongest and the eddy viscosity needs the aug-mentation provided by the near-wall stress model.

For these high Reynolds number simulations, wehave no DNS result to provide exact values of the 13SFS stresses for comparison, as was done by Gullbrandand Chow (2003). The turbulent stresses can, however,be compared to finer grid resolution simulationstreated as estimates of an exact solution. These esti-mates can then be used to calculate SFS stress values onthe coarser grid by using the definition of the turbulentstress tensor in Eq. (3). The finer grid data come fromARPS simulations of twice the resolution in each di-rection (83, 83, 83). The horizontal spacing is 16 m andthe vertical resolution is 5 m near the wall, andstretched above that to an average of 18.75 m over adomain of the same size as the base case (43, 43, 43)grid. The discrete quantities ũi can be obtained from theinstantaneous high-resolution fields a posteriori by us-ing a sharp cutoff filter that matches the coarse gridresolution (43, 43, 43) simulations, but we do not applythe cutoff filter here for simplicity (and the inability toapply it in the vertical direction); tests on the small-scale channel flow code showed that the cutoff filterapplied in the horizontal directions did not change themagnitude or shape of the resulting stress profiles sig-nificantly. We have also reconstructed the velocityfields to obtain ũ*I , which estimates the instantaneousvelocity fields from a twice-finer grid (163, 163, 163),though the reconstruction is especially limited near thewall. The high-resolution simulations also require a tur-bulence model; DRM-ADM0 is used instead of a stan-dard eddy-viscosity closure, because it yields goodmean velocity quantities (logarithmic near the wall).

Thus we calculate ̃ij � ũ*i ũ*j � ũ*i˜ ũ*j˜. The explicit filter

is applied at twice the (43, 43, 43) cell spacing (i.e., 4times the high-resolution spacing); a tophat filter is con-structed using the trapezoidal rule for the fine gridfields to match the filter width used in the LES simu-lations. The same filters are applied in the vertical di-rection. While this procedure provides only an estimateof the true SFS stresses, at the very least the compari-son shows how well we match highly resolved results(which are much more expensive to compute becauseof the 16-fold increase in computation time when thereduced time step is taken into account).

The turbulent stresses predicted by the coarse reso-lution simulations using dynamic reconstruction agreewell with the SFS stresses calculated from the finerresolution data. Results from DRM-ADM0 up toDRM-ADM10 are shown in Fig. 10 (without the near-wall stress contribution). The first point above the wallfor the fine grid stress is not shown because it cannot becalculated accurately; the boundary conditions affectthe application of multiple filters (especially in the ver-tical direction). In fact, the values of the extracted stresswithin 4�x of the wall are approximate because of thelimited resolution available. Stresses predicted by theDWL model (as well as Smagorinsky, not shown) aremuch too small. Increasing reconstruction on the coarse

FIG. 8. Comparison of uw SGS (thin lines) and RSFS (boldlines) stress profiles for the dynamic Wong–Lilly SGS model withincreasing RSFS reconstruction levels: DRM-ADM0; DRM-ADM1; and DRM-ADM5.

FIG. 9. Comparison of eddy-viscosity profiles for the dynamicWong–Lilly SGS model with increasing RSFS reconstruction lev-els, shown for the bottom 200 m of the domain: DWL, no recon-struction; DRM-ADM0; DRM-ADM1; and DRM-ADM5.

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grid improves agreement with the high-resolution re-sults. This shows that the reconstruction procedure con-tributes appropriately to the SFS stress, unlike themodels without reconstruction, which cannot capturethe RSFS motions. Because the reconstruction of theRSFS and the eddy-viscosity representation of the SGSstresses are imperfect, the near-wall stress model isneeded to contribute the necessary stress near the wall.

The optimal level of reconstruction is difficult to de-termine. Clearly, the increase in the RSFS stresses isnot linear in the reconstruction level. The series expan-sion models should converge to a fixed contribution.Figure 10 shows the increase between levels 0 and 1 ismuch larger than the difference between levels 5 and10. In the small-scale simulations of Gullbrand andChow (2003), the improvement between levels 5 and 10was also not very large. The higher reconstruction lev-els appear to converge, but the results may still differfrom the exact results because of SGS effects. It is alsopossible that numerical errors interact with the recon-struction procedure to limit our ability to capture all ofthe RSFS motions. Given the computation costs of re-construction (see section 5c3 below), it appears thatlevel-2 reconstruction is a good compromise. The in-creased cost from level 5 to 10 does not seem warrantedby the small difference in the results.

While the comparisons of mean velocity and � pro-files did not clearly distinguish among the simulations,these comparisons of second-order quantities show thatreconstruction of the RSFS stress is important. TheDWL results (without reconstruction) also producegood mean velocity profiles (Fig. 3), but cannot achievecorrect stress profiles. The stress profiles are especiallyimportant in predicting turbulent transport and mixing

of scalars, and it is only with the dynamic reconstructionclosure models that we are able to obtain accurate rep-resentations of these stresses.

4) EFFECTS OF GRID RESOLUTION AND ASPECTRATIO

A robust turbulence model should handle a variety ofdifferent grid configurations. Figure 11 shows the effectof doubling the grid resolution on the � profiles for theSmagorinsky and DRM-ADM0 simulations. The high-resolution grid uses twice as many points in each direc-tion over the same domain size to give a horizontalresolution of 16 m and �zmin of 5 m at the wall (seeTable 1). The near-wall stress model parameters werenot changed. The overshoot in the Smagorinsky simu-lations does not decrease; it simply moves closer to thewall. Similar behavior was observed for eddy-viscositymodels used by Mason and Thomson (1992) andKhanna and Brasseur (1997). The DRM-ADM0 showequally good results at both resolutions, giving in thiscase a somewhat grid-independent result for �.

The comparisons in Fig. 11 use a constant aspect ratioof �x/�z � 16/5 near the bottom wall. Larger aspectratios are common in LES applications where verticalgrid refinement is important for capturing the verticalstructure of the atmosphere (e.g., stratification). Aspectratios of 100:1 or larger are often used in mesoscalesimulations, sometimes with boundary layer parameter-izations. Such large aspect ratios distort the minimumresolvable eddies near the wall and place a large burdenon the SFS model which must compensate for this. Anaspect ratio of unity would be ideal (Kravchenko et al.1996), but such computations are prohibitively expen-sive. To better determine the contribution required ofthe near-wall stress model, we have examined aspectratios ranging from 1.0 to 12.8. Larger ratios are notfeasible on our small domain and lead to instabilities.

FIG. 10. Turbulent stresses computed a posteriori from velocityfields reconstructed from a fine resolution simulation [recon-structed from (83, 83, 83)], and the sum of RSFS and SGS stressescomputed by coarse grid simulations (43, 43, 43): DWL, DRM-ADM0, DRM-ADM1, DRM-ADM2, DRM-ADM5, and DRM-ADM10. The near-wall stress contribution is not included in thefigure. Averaged over 100 000 to 120 000 s.

FIG. 11. Comparison of nondimensional mean shear � profilesfor the Smagorinsky model and DRM-ADM0 for 433 and 833 gridsizes. Averaged from 75 000 to 100 000 s.


Figures 12a,b show the effects of a sample of fourdifferent grid spacings or aspect ratios on results usingthe Smagorinsky and DRM-ADM0 models. The near-wall stress contributions in the DRM-ADM0 simula-tions have been adjusted to give � profiles that areclose to unity (see Table 1). The required contributionof the near-wall stress model changes, as expected, be-cause this model was in part developed to accommo-date different grid aspect ratios near the wall.

The two independent near-wall stress layer param-eters are the layer height (hc) and the proportionalityfactor (Cc). We empirically determined the parametersneeded for good results over a range of grid sizes andaspect ratios. “Good results” mean the ability toachieve � profiles within 0.2 of unity near the wall.Tests indicate that the aspect ratio is less of a control-ling factor for the first parameter, the layer height, thanis the horizontal grid spacing. We argued earlier thatthe underresolved region at the wall extends between2�x (minimum resolved eddy size) and 4�x (minimumwell-resolved eddy size) from the wall. Most of the

simulations (of various grid sizes) give good results us-ing a near-wall stress layer height equal to 4�x. Thus, hcis different in the (83, 83, 83) versus the (43, 43, 43) gridcases, for example, for the same flow conditions, em-phasizing that the near-wall stress model is grid depen-dent. Some simulations give better results using layerheights of 2 or 3�x, as shown in Table 1. While thenear-wall stress layer thickness appears well deter-mined by this simple rule, there is likely a limiting fac-tor because the enhanced stress layer at coarse resolu-tions should not grow to extend beyond the boundarylayer depth.

Figure 13 shows the variation of Cc with the aspectratio �x/�z. The �z dependence of Cc is not strong, buta slightly better collapse of the data is obtained usingthe aspect ratio as opposed to �x. The value of Cccannot exceed unity, where the near-wall stress modelwould provide a stress at the first point above the wallexactly equal to the local wall stress. The required con-tribution of the near-wall stress model increases withaspect ratio (and also with �x), confirming our hypoth-esis that an enhanced stress layer is necessary to com-pensate for the underresolved region near the wall. Theneed for the near-wall stress layer does not disappearwhen the aspect ratio is near unity, but its influence isreduced. Numerous tests indicate that small deviationsfrom the values in Fig. 13 also give acceptable results.Therefore, Fig. 13 provides reasonable guidelines.While the near-wall stress model we have implementedprovides (with proper adjustment) the necessarysupplement to the total stress at the wall, this remainsan important topic for future research.

5) COMPUTATIONAL COST

The reconstruction models as well as the dynamiceddy-viscosity model are straightforward to implement,

FIG. 12. Comparison of nondimensional mean shear � profilesfor (a) Smagorinsky and (b) DRM-ADM0 for different grid spac-ings or aspect ratios: �x: �zmin � 32:10 (base case), 32:5, 64:10,128:10. Note that the axes are scaled differently. Averaged from75 000 to 100 000 s.

FIG. 13. Variation of near-wall stress model proportionalityconstant with grid aspect ratio. A curve indicating the trend of thedata is also shown.

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but the resulting increase in computational cost is notnegligible. We performed simple comparisons to deter-mine the computational costs of the new methods,though no effort was made to optimize the changes inthe turbulence modeling sections of the ARPS code.The base case simulations were performed using eightprocessors in distributed memory mode (via MPI) onan IBM SP Power4 system. The computation time testswere performed for 20 000 s.

Table 2 shows the increased computational cost ofvarious turbulence closures compared to the standardSmagorinsky model. From tests of DWL model withand without the near-wall stress model, we estimatethat the near-wall stress model adds approximately 5%to the total computation time. The reconstruction pro-cedure also adds significantly to the computation time.The increased cost of the turbulence model alone is alsoshown.

While the increase in cost is considerable, it must becompared with other means of obtaining equally accu-rate results. If, for example, the resolution weredoubled in each direction, the increase in computa-tional cost would be approximately 16 times, assumingthe time step were reduced by a factor of two. Evenwith this increase in resolution, Fig. 11 showed that theSmagorinsky model continues to overpredict the valuesof � near the wall. Given this alternative, the increasesin computational time of the DRM-ADM approachseem quite reasonable for the significant improvementin the results obtained. A compromise solution mightbe to use DRM-ADM0 up to DRM-ADM2, with anincrease in total computational cost of approximately50% over standard closure models. Finally, the in-creased cost of 50% seems reasonable when comparedto the factor of 2 increase in cost over static Smagorin-sky observed by Esau (2004) using the DMM, and the50% increase for using DMM over DSM observed byCui (1999), both in incompressible codes; the increasedcost of the turbulence model portion was not reported.Further optimization (in the filtering subroutines espe-cially) can make the code more efficient.

6. Summary and conclusions

The near-wall region in atmospheric boundary layerflow simulations is plagued by poor resolution and em-

pirically based wall models for the bottom stress, inaddition to SGS and RSFS turbulence modeling errors.This paper presented an approach to turbulence mod-eling over rough boundaries that incorporates the ideasof explicit filtering, velocity reconstruction, and near-wall stress modeling. The velocity partitioning ap-proach clearly delineates the required contributionsfrom each component. As a test case, we performedsimulations of the rotation-influenced neutral boundarylayer flow over flat terrain, similar to the case of An-dren et al. (1994).

Traditional turbulence closure models fail to producethe expected logarithmic region near the wall. Wetherefore began with the context provided by Carati etal. (2001), which led to insights about useful turbulencemodel decompositions. With explicit filtering, the totalSFS stresses are separated into RSFS and SGS compo-nents; the RSFS portion can be reconstructed, while theSGS portion must be modeled, though its required con-tribution is reduced because the RSFS stress is in-cluded. Even traditional closure models without ex-plicit filtering require reconstruction when finite-difference or finite-volume schemes are used. In thesecases, the lack of an explicitly defined filter makes itdifficult to determine a method for reconstruction. Ex-plicit filtering is also important for reducing numericalerrors from finite-difference schemes. Near the wall,the SGS contribution requires enhancement by a near-wall stress model to account for the effects of poorresolution, high grid-aspect ratios, and filter-inducedstresses at solid boundaries (Nakayama and Sakio 2004;Dubrulle et al. 2002). This framework aids understand-ing of the required contributions to subfilter-scalestresses, and simulation results show excellent agree-ment with similarity theory in the logarithmic region.

Two specific alternatives were presented for velocityreconstruction: the Taylor series expansion method ofKatopodes et al. (2000a) and Chow and Street (2002),and the approximate deconvolution method of Stolzand Adams (1999). The methods are equivalent withinthe order of accuracy of the expansions. Truncationerror from the Taylor series approach is kept to thedesired order of the filter width by explicitly neglectinghigher-order terms in the reconstruction of ij. TheADM procedure includes higher-order quantities in ij,so the truncation error is less well defined, but theADM provides a simple iterative method for calcula-tion of higher-order series expansions for reconstruc-tion, so it is used for the neutral boundary layer simu-lations in this work. Both methods reduce to the scale-similarity model of Bardina et al. (1983) at lowest orderand provide backscatter of energy from small to largescales. Both models also satisfy the full ij evolutionequations to a predefined order of accuracy (thoughthis is clearer when higher-order terms are neglected asdone when using the Taylor series approach).

While the series expansion approaches can be usedfor the RSFS component of the SFS stresses, the SGS

TABLE 2. Computational cost of different turbulence modelscompared to the Smagorinsky model.

ModelTurbulence model

cost factor Total cost factor

Smagorinsky 1.00 1.00DWL (no near wall) 2.33 1.17DWL 2.67 1.22DRM-ADM0 4.59 1.52DRM-ADM1 4.75 1.51DRM-ADM2 5.10 1.57DRM-ADM5 5.92 1.67DRM-ADM10 7.29 1.85


component must be modeled separately. We imple-mented the dynamic eddy-viscosity procedure of Wongand Lilly (1994), which requires only the test filterwidth as a free parameter. The dynamic Wong–Lillymodel has been used to provide necessary dissipation,without incurring the expected drawbacks of an eddy-viscosity model, because it is used in conjunction with ascale-similarity model. To enhance the near-wall stress,we use a model based on the canopy stress model ofBrown et al. (2001).

The combinations of the ADM reconstruction, theDWL, and the near-wall stress model significantly im-proved first- and second-order statistics. Mean velocityprofiles and nondimensional shear profiles (�) showedexcellent agreement with similarity theory in the loga-rithmic region. Examination of second-order statistics,particularly the uw stress decomposition, revealed thatincreasing reconstruction increases the contribution ofthe RSFS as well as the total SFS stresses, while corre-spondingly reducing the resolved uw stresses. This in-crease in the SFS stress clearly distinguishes the DRM-ADM models from the DWL and Smagorinsky resultswithout reconstruction. Using finer resolution data toreconstruct a high-resolution velocity field for compari-son, we showed that the increased SFS stresses (fromsimulations at the base case low resolution) approachthe required SFS stress predicted by the finer resolutiondata. The DWL and Smagorinsky results strongly un-derpredict stresses. These observations are in agree-ment with those of Gullbrand and Chow (2003) whoshowed that the stresses from small-scale channel flowsusing the DRM approached the predicted stresses fromDNS data. Accurate prediction of stresses is especiallyimportant for atmospheric flows, where turbulenttransport and mixing of scalars such as temperature,moisture, aerosols, and pollutants are of interest.

These results for high Reynolds number atmosphericflow simulations over rough boundaries using explicitfiltering and reconstruction are very encouraging. Sev-eral different grid resolutions and grid aspect ratioshave been tested, and reasonable results have been ob-tained by accounting for the required variation in thenear-wall stress model as a function of horizontal gridspacing and grid aspect ratio. We found that the near-wall stress model alone is insufficient for correctinglarge-eddy simulations near the lower boundary; itmust be used in conjunction with a sophisticated ex-plicit filtering and reconstruction approach so that itprovides only the necessary near-wall SGS stress. Theimprovements obtained from the near-wall stressmodel point to the need for more sophisticated SGSclosures that account for the effects of a rough bound-ary at the lower wall.

In summary, the steps required to achieve improvedneutral boundary layer simulations are as follows. First,an explicit filter width must be chosen that is compat-ible with the discretization scheme used in the code

(Chow and Moin 2003; Ghosal 1996). We chose theexplicit filter to be twice the grid size, in agreement withthe fourth-order central differencing used. The explicitfilter is the basis for the reconstruction procedure,which is easiest to implement using the ADM ap-proach. Level-0 reconstruction (DRM-ADM0) signifi-cantly improves the mean quantities, and higher-orderreconstruction further improves the representation ofthe SFS stresses especially. The reconstruction ap-proach is used with a dynamic eddy-viscosity procedurethat requires a test filter, typically taken to be twice theexplicit filter. In our case, the test filter width was 4times the grid spacing. The test filter is used to calculateRSFS and SGS contributions at the test level; they arethen compared to their values at the explicit filter levelto determine the appropriate eddy-viscosity coefficient.Finally, the SGS contribution from the dynamic eddy-viscosity model is not sufficient near the wall, so a sepa-rate near-wall stress model is added to enhance thestress. The thickness of this near-wall stress layer istypically chosen to be between 2 and 4�x. The propor-tionality factor that determines the contribution of theenhanced near-wall stresses varies between 0.4 and 0.9,depending primarily on the grid aspect ratio. Thus, thecombined RSFS, SGS, and near-wall stress approachrequires two parameters for the enhanced near-wallstress, plus a priori selection of the explicit and testfilter widths.

As LES is applied to problems where more of theenergy of the flow is unresolved, the accuracy of theSFS model becomes increasingly important. Althoughthere is increased computational cost, the explicit fil-tering and RSFS approach is warranted by the resultingimprovements in the simulation results. To our knowl-edge, we have demonstrated for the first time thatscale-similarity or reconstruction models used with ex-plicit filtering can dramatically improve results for highReynolds number, rough, atmospheric boundary layerflows. While we have selected specific RSFS and SGSmodel components, the velocity partitioning approachcan be used with other, simpler SGS closures (e.g., non-dynamic eddy-viscosity models) to improve simulationresults. A low-order RSFS component can easily beadded to any code with zero-order reconstruction (theBardina term); this will improve total SFS stress repre-sentations when the RSFS component is used with anexisting eddy-viscosity model. Further research is re-quired to determine optimal SGS and near-wall stressrepresentations, and the RSFS/SGS approach shouldalso be tested for other flow conditions. The RSFS com-ponent (obtained by series expansions) can remain un-changed, but the SGS model may need adjustment de-pending on the flow and grid resolution. Preliminarytests on moderately convective boundary layer (Chow2004, appendix F) and moderately stable boundarylayer simulations also show improved agreement withsimilarity theory.

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Acknowledgments. The support of a National De-fense Science and Engineering Graduate fellowship(FKC) and NSF Grants ATM-0073395 (Physical Me-teorology Program: W. A. Cooper, Program Director)(FKC and RLS), ATM-9909007 and 0129892 (MX) aregratefully acknowledged. Acknowledgment is alsomade to the National Center for Atmospheric Re-search, which is sponsored by NSF, for the computingtime used in this research. Part of this work was per-formed under the auspices of the U.S. Department ofEnergy by the University of California, Lawrence Liv-ermore National Laboratory under Contract W-7405-Eng-48.

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