EXPLICIT FILTERING AND RECONSTRUCTION TURBULENCE MODELING FOR LARGE-EDDY SIMULATIONS OF FIELD-SCALE FLOWS

Transcription

1 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 1 EXPLICIT FILTERING AND RECONSTRUCTION TURBULENCE MODELING FOR LARGE-EDDY SIMULATIONS OF FIELD-SCALE FLOWS Fotn Katopodes Chow 1 and Robert L. Street 2 ABSTRACT Large-eddy smulaton (LES) has become a wdely-used method for computaton of feld-scale envronmental flows; however, challenges reman n determnng the optmal turbulence closure method. Recently, the concept of explct flterng and reconstructon has become recognzed as a means to obtan more accurate smulaton results (Carat et al. 2001, Wnckelmans et al. 2001, Stolz et al. 2001, Gullbrand and Chow 2003). In ths paper we revew these new methods and extend them to flow over a rough boundary at feld scale. By parttonng the subflter-scale total turbulent stress nto resolved and unresolved components, t becomes clear that the resolved component can be reconstructed usng a seres expanson to approxmate the nverse flter operaton. The unresolved stress,.e., subgrd-scale stress, must be modeled. We apply the dynamc reconstructon model (DRM), usng reconstructon plus a dynamc eddy vscosty model and a near-wall enhanced stress model, to smulate an ncompressble flud flow n a channel over a rough boundary. Standard eddy vscosty closures fal to reproduce the expected logarthmc velocty profle when used alone. The DRM produces much mproved results, yeldng a logarthmc profle that extends over 10-15% of the boundary layer depth wth accompanyng nondmensonal shear numbers close to unty. 1. INTRODUCTION Flows n coastal waters, estuares, rvers, and lakes, and over the surface of the earth, are strongly nfluenced by turbulent flud motons. Turbulence affects the rate of transport of momentum, heat, salnty, sedments, and other scalar quanttes such as nutrents and contamnants. Numercal smulatons are currently the only practcal method to obtan fully three-dmensonal realzatons of such complex envronmental flows. Statstcally steady-state modelng tools such as Reynoldsaveraged Naver-Stokes (RANS) smulatons and low-frequency smulatons obtaned by tme averagng over fnte perods have greatly advanced the applcaton of numercal models to envronmental flows (Rod 1995). The advent of faster computers and mproved numercal methods, however, now makes an alternatve approach, that of large-eddy smulaton (LES), feasble. LES provdes an unsteady, fully tme-dependent and three-dmensonal soluton able to capture the dynamcs of evolvng turbulent envronmental flows. The accuracy of such smulatons s dependent 1 Research Assstant (Ph.D.), Envronmental Flud Mechancs Laboratory, Dept. of Cvl and Envronmental Engneerng, Stanford Unversty, Stanford CA , USA stanfordalumn.org) 2 Wllam Alden and Martha Campbell Professor n the School of Engneerng, Envronmental Flud Mechancs Laboratory, Dept. of Cvl and Envronmental Engneerng, Stanford Unversty, Stanford CA , USA stanford.edu)

2 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 2 on the grd resoluton, dscretzaton schemes, and other numercal parameterzatons, especally the turbulence closure model and the treatment of the stress nduced by rough boundares. Determnng the optmal turbulence closure method remans a challenge for LES as for RANS; however, LES seeks to resolve as many turbulence scales as possble, reducng the range of remanng flud motons that must be modeled, as opposed to RANS n whch typcally all of turbulent scales are modeled. Recently, understandng of the nteracton of flters and numercal schemes has mproved the approach to formulatng closure models for LES. Our goal, then, s to set a context for and to descrbe a systematc and accurate approach for modelng n LES. In LES, the flud feld s spatally fltered to separate large eddes from smaller motons; the larger scales of the turbulent flow are explctly smulated, whle the effect of the smaller, subflter, scales on the large scales s modeled. The presence of a numercal grd subdvdes the subflter-scale (SFS) motons nto resolved and unresolved portons. The resolved subflter-scale () motons can be reconstructed usng a scale-smlarty approach, whle the unresolved subflter-scale (USFS) motons (also called subgrd-scale (SGS)) must be modeled (Carat et al. 2001). Both the and SGS models must be based on knowledge of the resolved-scale behavor alone. The parttonng of SFS motons nto and SGS facltates an understandng of the roles of varous turbulence model components. Reconstructon modelng of the subflter-scale (SFS) stresses requres the defnton and applcaton of an explct flter (larger than the grd spacng) n the LES computaton. In contrast, tradtonal LES treats the grd as an mplct flter operaton, but the nature of the flter s unknown, makng reconstructon dffcult. Explct flterng and reconstructon are especally useful for reducng numercal errors n the context of fnte volume or fnte dfference codes, whch are used n envronmental applcatons; spectral methods are not useful for flows over complex geometres and do not requre reconstructon (Wnckelmans and Jeanmart 2001). Gullbrand and Chow (2003) presented small-scale (low Reynolds number) turbulent channel flow results that showed that explct flterng and reconstructon methods (the dynamc reconstructon model (DRM)) have the potental to reduce numercal errors n fnte volume formulatons of LES. In ths paper, we use a smlar approach for smulatons of feld-scale (hgh Reynolds number) flow. The seemngly small dstncton of the presence of ether a smooth or a rough wall actually has very large consequences on the numercal results. A rough bottom boundary requres the use of approxmate boundary condtons (e.g., specfyng a log law at the wall) whch ntroduce further uncertanty nto the smulatons. We ntroduce three-dmensonal flters so that our approach s general enough to be appled to flow over complex boundares, e.g., the coastal ocean, rver and estuary beds and natural terran. For smplcty and clarty we descrbe a channel flow where the flat bottom boundary s rough and the top boundary s free slp (see also Chow and Street 2002); the method has been successfully appled to complex boundares as well (Chow, 2004). Standard turbulence closures for feld-scale flows use eddy vscosty models and hence gnore the contrbuton of the resolved subflter-scale stresses. These eddy vscosty closures are unable to produce the expected logarthmc regon near the wall n open channel flows. Our approach s to use reconstructon to mprove the representaton of the resolved subflter-scale () stresses (Gullbrand and Chow 2003, Stolz et al. 2001), and a dynamc eddy vscosty model for the subgrdscale (SGS) stresses (Wong and Llly 1994). Combnng reconstructon and dynamc eddy vscosty models (DRM) yelds a sophstcated (and hgher-order) verson of the well-known mxed model of Bardna et al. (1983); the explct flterng and reconstructon procedures delneate clearly the contrbuton of the and SGS motons. To better represent the rough lower boundary, we mplement a near-wall stress model to account for the stress nduced by flterng near a sold boundary as well as for the effect of the grd aspect rato. In secton 2, we dscuss the deas behnd the parttonng of and SGS motons; n secton 3, reconstructon and SGS modelng approaches are descrbed. Secton 4 gves results for large-eddy smulatons of flow over a flat rough boundary.

3 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 3 2. SUBFILTER-SCALE PARTITIONING To facltate understandng of the requrements n SFS modelng and especally to mprove turbulence models n the near wall regon, t s useful to consder velocty parttonng schemes such as those of Carat et al. (2001), Zhou et al. (2001), and Hughes (2001). Fgure 1 shows a schematc of a typcal energy spectrum from a turbulent flow. The parttonng s based on the applcaton of a spatal flter (whch s smooth n wave space) n addton to a dscretzaton operator (approxmated by a spectral cutoff flter) needed to solve the LES equatons on a dscrete grd. The spectrum can thus be separated nto three parts. The low wavenumber porton s fltered and well-resolved on the grd, and s contaned n the velocty u ~, where the tlde operator represents the dscretzaton and the overbar an explct smooth flter. The mddle porton (shaded) represents subflter-scale motons that are between the flter and grd cutoffs and hence resolvable on the grd. These resolved subflterscale motons can theoretcally be reconstructed by an nverse flter operaton. However, reconstructon s lmted by numercal errors (NE) whch ncrease near the grd cutoff (due to the modfed wavenumber effect). The last porton (to the rght of the vertcal dashed lne) contans subgrd-scale motons that cannot be resolved on the grd and must be modeled. Fgure 1 Schematc of velocty energy spectrum showng parttonng nto resolved, subflter-scale, and subgrd-scale motons. The numercal error regon s also shown. The grd cutoff s ndcated by the vertcal dashed lne at wavenumber k g =π/ g (correspondng to the mnmum resolvable wavelength), and the flter by the curved dashed lne. The explct flter functon G of wdth f s appled to a flow varable f wth f ( x,, t) = G( x, x', ) f ( x', t) dx' (1) f The tophat flter s commonly used n LES and s defned by G = 1 for x' < f / 2 and zero otherwse. (Applcaton of the trapezodal rule yelds a dscrete verson of ths flter whch can easly be appled numercally; n one dmenson, f m = 0.25 f m f m f m+ 1, for f = 2 g, where m s the ndex of the dscrete varable, and g s the grd spacng.) Applyng ths spatal flter and the dscretzaton operator to the Naver-Stokes and contnuty equatons yelds the computatonal LES equatons as f

4 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 4 u~ t u~ u ~ + x 1 ~ p = ρ x ~ τ x x u~ ; = 0, (2) x where vscous terms have been neglected. Here u ~ are the fltered (and resolved) velocty components, ~ p the pressure, and ρ the densty. It s assumed that the flterng operaton commutes wth the spatal dervatves. We defne the total SFS stress as τ = u u u ~ u~. (3) Note that when the SFS stress appears n the LES equatons (eq. 2) the addtonal tlde ndcates the effect of the dscretzaton operator (whch s also a de-alasng step). By use of the above parttonng deas, the full turbulent stress can be decomposed nto resolved and unresolved portons: τ = u u u~ u~ = ( u u u~ u~ ) + ( u ~ u ~ u~ u~ ) (4) SFS The frst par of terms on the RHS are the subgrd-scale stresses, τ SGS, whch depend on scales beyond the resoluton doman of the LES; they contan the unclosed nonlnear term, u u, whch must be modeled. The last par of terms are the fltered-scale stress porton, τ, whch depends on the dfferences between the exact and fltered velocty felds wthn the resoluton doman. Ths resolved subflter-scale component, τ, can theoretcally be reconstructed because t s a functon of u ~ whch can be obtaned by a deconvoluton procedure (descrbed below). Note that n a contnuous doman, an nfnte expanson n a seres model for τ would gve an exact soluton (Katopodes et al. 2000a), snce there would be no contrbuton from subgrd-scale effects. In a dscrete doman, the contrbuton of the totalτ SFS, and thus τ SGS, ncreases wth decreasng grd resoluton. SGS models that are currently employed may not be very good representatves of the true SGS motons. It s hoped that by reconstructng the stress, the overall representaton of the SFS stress wll be mproved, as the SGS contrbuton wll decrease overall. However, near a rough wall, eddy szes decrease much faster than any possble grd stretchng, and the bulk of the stress contrbuton comes from SGS terms (Sullvan et al. 2003). In general, the total stress s gven by τtotal = τresolved + τ + τsgs (5) Because eddes scale roughly as dstance from the boundary and the flter and grd cutoffs are fxed, as we approach the wall (z 0), t must be thatτ Resolved 0 andτ 0 because there are no eddes n ther respectve spectral areas and all of the eddes are subgrd. Thus, τ total τ SGS as z 0. The stress on the wall s gven by a wall model such as the log law. Thusτ SGS supports the total stress at the wall, suggestng the use of a specfc near-wall stress model whch represents the stress nduced by the rough boundary. τ SGS also exsts throughout the flow, but t may be small away from the wall, dependng on the grd dscretzaton and the turbulent processes occurrng n the flow. There could perhaps be a unfed approach to SGS τ over the whole doman, but we consder separate near-wall stress and general SGS models n ths work.

5 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 5 3. SUBFILTER-SCALE RECONSTRUCTION AND SUBGRID-SCALE MODELING Usng the above framework for the turbulent stress, we can construct models for the and SGS components separately. We frst focus on the components whch can be reconstructed. 3.1 Reconstructon Several methods have been proposed to represent such subflter-scale motons. Bardna et al. (1983) made a semnal contrbuton by ntroducng the scale-smlarty model. Scale-smlarty models create an approxmaton to the full velocty feld and thus estmate the stress. In Bardna's model, the dscrete full velocty s approxmated by the fltered velocty, u ~ u~, to obtan τ u ~ u ~ u~ u ~. Ths was the frst model that used the smallest resolved scales as ts bass. Later models have ncluded those of Yeo et al. (1988), Shah and Ferzger (1995), Geurts (1997), Stolz et al. (2001), Zhou et al. (2001), and Dubrulle et al. (2002) (see Domaradzk and Adams (2002) for an excellent revew). For the stress termτ, whch can be expressed n terms of the resolved velocty, we have frst mplemented the seres model of Katopodes et al. (2000a, 2000b). Ths model uses successve nverson of a Taylor seres expanson to express the unfltered velocty n terms of the fltered velocty. If the flter s sotropc, the expanson reduces to 2 ~ ~ f 2~ 4 u = u u + O( ) (6) 24 whch s second order n the flter wdth, f. The expanson can be extended to an arbtrary order of accuracy by ncludng more terms n the seres, though these can become cumbersome to compute. The approach can smlarly be appled to the scalar transport equaton, as done by Katopodes et al. (2000b). An ansotropc tophat flter s used n the smulatons, though an sotropc flter s shown here for smplcty. Other spatally compact flters gve smlar results, wth a change n the expanson coeffcents. Reconstructon of the stress tensor can also be acheved by the teratve deconvoluton method of van Cttert (1931). The unfltered quanttes can be derved by a seres of successve flterng operatons wth u ~ = u~ + ( I G) * u ~ + ( I G) *(( I G) * u ~ ) +L (7) where I s the dentty matrx, and G s the explct flter. Level-n reconstructon ncludes the frst n+1 terms of the seres. Ths reconstructon s used by Stolz et al. (2001) who call the model the approxmate deconvoluton model (ADM). Ths expanson can also be extended to an arbtrary order of accuracy by ncludng more terms n the seres, though t s not mmedately obvous what the order of magntude of the next set of terms s. Computaton of hgher-order terms s straghtforward, as t smply requres repeated applcaton of the same flter operator. The recursve Taylor seres and ADM approaches are equvalent to a gven truncaton error (see Stolz et al. (2001) and Chow (2004)). The use of the Taylor seres expansons makes t easer to preserve the desred order of the reconstructon; however the ADM approach s much smpler to

6 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 6 mplement numercally. We derve models for τ by substtutng the seres expanson for the reconstructed velocty ( u ~ ) drectly nto the defnton to obtan τ u ~ ~ ~ ~. (8) = u u u For the ADM approach, nothng further s requred. In the Taylor seres approach, we expand ths expresson to derve models of the specfed order of accuracy n the flter wdth (see Chow and Street 2002). It s mportant to note that no parameters occur n ether model except the choce of flter wdth and the number of terms to keep n each seres. No assumptons are made about the form of the motons. The models are thus able to capture ansotropc motons better than eddy vscosty models. Both of these seres expansons reduce to the Bardna scale-smlarty model at lowest order. In addton to havng desrable scale-smlarty propertes, the evoluton equaton that can be developed (Katopodes et al. 2000a) for the approxmate τ ndcates that these resolved subflterscale stresses are nfluenced by buoyancy and Corols forces, as well as dffuson, pressure and advecton terms, ust as the resolved veloctes are. Thus the expresson (eq. 8) for τ captures the effects of all of the relevant physcal mechansms, to a gven order n the flter wdth. Further detals on the propertes of the reconstructon procedure can be found n Gullbrand and Chow (2003), Stolz et al. (2001), and Chow (2004). The Taylor seres and ADM approaches produce essentally the same results at the gven truncaton level; therefore only ADM results are presented below. 3.2 Subgrd-Scale Modelng The problem of representng the stresses has thus essentally been solved by usng seres expansons. Asde from the ssues that arse because of numercal errors due to dscretzaton schemes n the numercal model (as well as n the reconstructon procedure), the reconstructon procedures descrbed above are exact up to the truncaton error. Unfortunately, the turbulence closure problem remans n the SGS terms (see eq. 4; the unclosed term u u s n the SGS porton of the total SFS stress). For lack of a better framework, a smple gradent dffuson form s used here for modelng the unclosed term, as suggested by Carat et al. (2001): τ = 2ν (9) SGS T S ~ where ν T s the eddy vscosty, and S ~ s the resolved stran rate tensor. Despte the known shortcomngs of ths model, t s convenent to use when energy transfer to the subgrd scales s desred. A common treatment n LES s to use the Smagornsky model (1963), whch assumes ~ ~ 2 1/ 2 ν T = ( CS g ) (2SS ) (10) where C S s the Smagornsky coeffcent. We have also chosen to mplement the dynamc model of Wong and Llly (1994); here the eddy vscosty s determned dynamcally n space and tme. Such gradent dffuson models are often appled to represent the entre stress tensor, τ, whereas here we apply the eddy vscosty model as part of a mxed model, so ts contrbuton wll not be as

7 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 7 pronounced (see Zang et al. 1993). We use the ADM together wth the DWL n a dynamc procedure to obtan the total SFS stress ~ τ = S (11) u~ u~ u~ u~ 4/ 3 2Cε whch we call the dynamc reconstructon model (DRM) (smlar to Gullbrand and Chow 2003). The dynamc procedure ncludes the nfluence of the contrbuton when calculatng the eddy vscosty coeffcent, C ε, usng the procedure for mxed models of Zang et al. (1993) and Vreman et al. (1994) (where an extra test flter must be defned for the dynamc model). Reconstructon seres from levels zero up to fve are used here and they are denoted DRM-ADM0 through DRM-ADM5. Note that DRM-ADM0 s the same as the dynamc mxed model (DMM) usng explct flterng as mplemented by Vreman et al. (1994) (except that we use the DWL nstead of DSM). 3.3 Near-wall Enhanced Stress Model The resoluton lmtatons present when smulatng a feld-scale flow requre specal treatment for the turbulence model near the rough lower boundary. We have found that any combnaton of and eddy vscosty models has lmtatons near the sold lower boundary, where eddy szes decrease much more rapdly than the grd spacng. The vertcal grd spacng s nvarably smaller than the horzontal one, but eddes are generally unform n sze; whle they may be well-resolved n the vertcal, they are not n the horzontal drectons, hence ntroducng errors. Because 2 x s the mnmum eddy sze resolved n the horzontal (arguably 4 x s the smallest well-resolved eddy), over a vertcal dstance of 2 x, eddes of ths sze are stll under-resolved. Ths lack of resoluton mples that an addtonal stress term may be needed near the wall to represent these motons. In addton to the nfluence of the grd aspect rato, the physcal exstence of subgrd roughness may alter the dstrbuton of stresses near the surface. Nakayama and Sako (2002) examned the effects of subgrd roughness by performng a drect numercal smulaton (DNS) of flow over a wavy bottom boundary that conssted of small and large wavelengths. By flterng the DNS flow feld and the wavy DNS boundary, they obtaned a snapshot of an deal LES soluton over the large wavelength boundary wth subgrd roughness. The fltered veloctes at the surface n the LES doman were now apparently only nfluenced by the larger wavelength topography, but the subflterscale roughness elements (the smaller wavelengths n the orgnal DNS boundary) had generated extra stress near the new, smoother boundary. Ths smple test showed that the surface stress, orgnally dstrbuted over the DNS boundary, became dstrbuted over a fnte vertcal thckness above the smoothed LES boundary. Smlar to Nakayama and Sako (2002), Dubrulle et al. (2002) (see ther Eq. 16) also found that flterng near a sold boundary generates extra stress terms. Motvated by the above arguments, we follow the method of Brown et al. (2001) to create an enhanced stress model whch serves to dstrbute stresses generated at the rough surface over a fnte dstance from the wall. The model can be expressed as a forcng term n the horzontal momentum equatons, C a( z) u~ u~, where k = 1, 2. Here C c s a scalng factor and a(z) s a constant smoothng c k 2 functon; both are pre-determned. We use a( z) = cos ( πz / 2h c ) for z < h c, where h c s the heght over whch the enhanced stress has nfluence. When mplemented numercally, the enhanced stress s treated as part of the turbulence closure stress term, and therefore s ntegrated numercally usng the trapezodal rule from τ C a( z) u~ u~ dz (12) k, near wall = c k

8 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 8 where the ntegraton constants are chosen so that τ = 0 at z = h k, near wall c. Ths stress s then drectly added to the τ k3 terms contrbuted from the other SFS model components. Brown et al. (2001) determned the coeffcents C c and h c descrbng the strength and extent of ther enhanced-stress forcng by matchng smulaton results to expermental data. Cederwall (2001) also mplemented the model to mprove representaton of the rough bottom boundary condton; C c was selected such that the near-wall stress model augmented the total stress at the frst grd pont above the wall to make t equal to the total local bottom shear stress. Instead we allow C c to be locally proportonal to the bottom shear stress n each horzontal drecton. The proportonalty factor s chosen to allow the near-wall stress model to provde the necessary augmentaton that wll yeld logarthmc mean velocty profles near the wall. The vertcal extent of the stress layer, h c, s chosen to be 4 x, as motvated by the arguments above. The coeffcents do vary wth grd spacng and aspect rato as expected (Chow 2004). 4. FIELD-SCALE LARGE-EDDY SIMULATIONS To llustrate the /SGS approach n smulatons of feld-scale envronmental flows, we have smulated a smple channel flow at hgh Reynolds number and wth a rough bottom boundary. Smlar flows were consdered by Andren et al. (1994) and Porté-Agel et al. (2000), among others. Our /SGS approach has been mplemented n the Advanced Regonal Predcton System (ARPS) (Xue et al. 2000) and appled to solve the ncompressble, fltered Naver-Stokes equatons (Xu et al. 1996). ARPS was orgnally desgned as a compressble, mesoscale meteorologcal largeeddy smulaton code and we do use t n that form for valley wnd smulatons, for example (Chow 2004). Here, the boundary layer flow s drven by a constant pressure gradent (n the x drecton) whch at steady-state gves a unque value of u * (the frcton velocty, ~ 0.55 m/s) and lnear total uw stress profles. The grd sze s 43 3 wth a resoluton of 32 m 32 m n the horzontal. In ARPS ths corresponds to a doman sze of x (n-3) = 1280 m n each horzontal drecton. In the vertcal, a stretched grd s used, wth 10 m spacng near the bottom and up to 65 m near the top of the doman, gvng an average spacng of 37.5 m, and a doman heght of 1500 m. The no-slp condton cannot be appled at the bottom boundary because of nsuffcent near-wall resoluton. Hence, the top and bottom boundares are treated as rgd free-slp boundares, and surface fluxes are parameterzed to account for the nfluence of the rough bottom surface. The ARPS code parameterzes the momentum fluxes at the surface by applyng an nstantaneous logarthmc drag law (used here wth constant drag coeffcents) at each grd pont. The bottom roughness s set to 0.1 m and the drag coeffcent s derved from ths by applyng the logarthmc velocty condton to the frst grd cell above the wall (at heght 0.5 z mn ). At the lateral boundares, perodc condtons are used. Smulatons were run wth a 0.5 s large tmestep and 0.05 s small tmestep (for the mode-splttng scheme). Hgher and lower grd resolutons as well as dfferent grd aspect ratos were also studed (the results are not presented here; see Chow (2004)). To obtan good statstcs, the smulatons were run to s and results were averaged over the last s. The flow n the control case (usng the Smagornsky closure) s ntally unform and small perturbatons n the velocty feld trgger nstabltes n the flow untl t becomes fully turbulent. All other runs are ntalzed usng the s restart fle from the control case. We compare the results of smulatons usng several dfferent turbulence models. For the control case, we use the standard Smagornsky model wth C S = Fgure 2a shows the mean velocty from smulatons usng the Smagornsky model, the dynamc Wong-Llly model, and for the DRM-ADM5 hybrd model (level-5 reconstructon, DWL, and near-wall stress). The logarthmc regon s expected to extend over 10-15% of the boundary layer depth. It s clear that the tradtonal

9 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 9 approach usng the Smagornsky model sgnfcantly over-predcts the mean velocty profles, whle the models usng explct flterng provde sgnfcant mprovement. All profles are for horzontally planar-averaged and tme-averaged quanttes. Fgure 2 Comparson of (a) mean velocty and (b) non-dmensonal shear Φ profles for the Smagornsky model, the dynamc Wong-Llly model, and for DRM-ADM0, 1, and 5. A more senstve measure of a model's performance s the non-dmensonal shear gradent, Φ, whch s defned as 2 κz < u > Φ =, (13) u* z where <u> s the horzontally planar averaged velocty n the x drecton, and κ s the von Karman constant. Φ profles should be near unty n the logarthmc regon n the bottom 10-15% of the boundary layer. Fgure 2b shows that tradtonal eddy vscosty closures such as the Smagornsky model provde excessve shear, gvng Φ values of 1.7 near the wall; consequently, the mean velocty s over-predcted. The profles have been smoothed to remove 2 z waves. The DWL results show that the strong overshoot from the statc Smagornsky model s corrected by usng the dynamc eddy vscosty model together wth the enhanced near-wall stress. Whle the DWL results are good qute close to the wall, the log regon extends only to about 150 m. For neutrally-stratfed flows, the logarthmc regon s expected to extend further than 10% of the boundary layer depth (Sullvan et al. 1994). When the reconstructon and dynamc eddy vscosty models are used together wth the near-wall stress term (DRM-ADM), values of Φ wthn 0.1 or better of the deal (unty) are obtaned. Increasng the level of reconstructon further mproves the results, and extends the logarthmc regon to about 20% of the boundary layer depth. Increasng reconstructon also decreases the devaton from unty close to the wall. Profles n fgure 3a show the parttonng of the normalzed uw stress for DRM-ADM0. The contrbuton of the stresses decays near the wall as expected, where SGS stresses become ncreasngly mportant (see Secton 2). Increasng reconstructon levels ncreases the (and total SFS) stresses, as seen n Fgure 3b; ths s consstent wth the fndngs of Gullbrand and Chow (2003). Comparsons to hgher resoluton results ndcate that the SFS stresses predcted by eddy vscosty models are strongly under predcted (further detals are n Chow (2004) and Gullbrand and Chow (2003)).

10 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 10 Fgure 3 (a) Dstrbuton of stresses for the DRM-ADM0 results and (b) comparson of uw resolved (thn lnes) and SFS (bold lnes) stress profles for Smagornsky, DWL, DRM-ADM0, and DRM- ADM5. 5. CONCLUSIONS We have presented an explct flterng and reconstructon approach for large-eddy smulatons over rough boundares. By usng seres expansons to reconstruct the resolved subflter-scale stress, the overall representaton of the subflter-scale stress mproves. The subgrd-scale stress must be modeled, but the relatve contrbuton of ths component s reduced. Explct flterng also helps to reduce numercal errors that arse n fnte dfference formulatons. Results for boundary layer flow over a rough wall usng the DRM wth a near-wall stress model show excellent agreement wth smlarty theory logarthmc velocty profles, a sgnfcant mprovement over standard eddy vscosty closures. Thus, the steps requred to acheve mproved turbulent flow smulatons are as follows. Frst, an explct flter wdth must be chosen whch s compatble wth the dscretzaton scheme used n the code. Here we have chosen the explct flter to be twce the grd sze. Ths explct flter forms the bass for the reconstructon procedure, whch s easest to mplement usng the ADM approach. Level-0 reconstructon (DRM-ADM0) already provdes a sgnfcant mprovement n the mean quanttes; hgher order reconstructon further mproves the representaton of the SFS stresses n partcular. The reconstructon approach s used together wth a dynamc eddy vscosty procedure; ths requres the defnton of a test flter, typcally taken to be twce the explct flter. (In our case, ths results n a test flter wdth equal to four tmes the grd spacng.) Fnally, as the SGS contrbuton s not suffcent near the wall, a separate near-wall stress model s added to enhance the stress. The heght of ths near-wall stress layer s typcally chosen to be 4 x. The proportonalty factor whch determnes the magntude of the contrbuton of the enhanced near-wall stresses s found to vary wth the grd aspect rato. Thus, the combned, SGS and near-wall stress approach requres two parameters for the enhanced near-wall stress, n addton to a pror selecton of the explct and test flter wdths. As LES begns to be appled to problems n whch more of the energy of the flow s unresolved, the accuracy of the SFS model becomes ncreasngly mportant. We have demonstrated that scale-smlarty or reconstructon models used wth explct flterng can dramatcally mprove results for hgh Reynolds number, rough boundary layer flows.

11 ADVANCES IN HYDRO-SCIENCE AND ENGINEERING, VOLUME VI 11 ACKNOWLEDGEMENTS The support of a Natonal Defense Scence and Engneerng Graduate fellowshp [FKC] and NSF Grant ATM (Physcal Meteorology Program: W.A. Cooper, Program Drector) [FKC and RLS] s gratefully acknowledged. Acknowledgement s also made to the Natonal Center for Atmospherc Research, whch s sponsored by NSF, for the computng tme used n ths research. REFERENCES Andren, A., Brown, A., Graf, J., Mason, P., Moeng, C.-H., Neuwstadt, F., and Schumann, U. (1994) Large-Eddy Smulaton of a Neutrally Stratfed Boundary Layer: A Comparson of Four Computer Codes, Quarterly Journal of the Royal Meteorologcal Socety, Vol. 120, pp Bardna, J., Ferzger, J. and Reynolds, W. (1983) Improved Turbulence Models Based on Large Eddy Smulaton of Homogeneous, Incompressble, Turbulent Flows, Techncal Report TF- 19. Department of Mechancal Engneerng, Stanford Unversty, Stanford, Calforna. Brown, A., Hobson, J.M. and Wood, N. (2001) Large-Eddy Smulaton of Neutral Turbulent Flow Over Rough Snusodal Rdges, Boundary-Layer Meteorology, Vol. 98, pp Carat, D., Wnckelmans, G. and Jeanmart, H. (2001) On the Modellng of the Subgrd-Scale and Fltered-Scale Stress Tensors n Large-Eddy Smulaton, Journal of Flud Mechancs, Vol. 441, pp Cederwall, R. (2001) Large-Eddy Smulaton of the Evolvng Stable Boundary Layer Over Flat Terran. Ph.D. Dssertaton, Stanford Unversty. Chow, F.K. (2004) Subflter-Scale Modelng for Large-Eddy Smulatons of the Atmospherc Boundary Layer Over Complex Terran. Ph.D. Dssertaton (to be submtted), Stanford Unversty. Chow, F.K. and Street, R.L. (2002) Modelng Unresolved Motons n LES of Feld-Scale Flows, 15th Symposum on Boundary Layers and Turbulence, Amercan Meteorologcal Socety, pp Domaradzk, J. A. and Adams, N. A. (2002) Drect Modellng of Subgrd Scales of Turbulence n Large Eddy Smulatons, Journal of Turbulence, Vol. 3, Art. No. 24. Dubrulle, B., Laval, J.-P., Sullvan, P.P., and Werne, J. (2002) A New Dynamcal Subgrd Model for the Planetary Surface Layer. Part I: The Model and A Pror Tests, Journal of Atmospherc Scences, Vol. 59, pp Geurts, B.J. (1997) Inverse Modelng for Large-Eddy Smulaton, Physcs of Fluds, Vol. 9, pp Gullbrand, J. and Chow, F.K. (2003) The Effect of Numercal Errors and Turbulence Models n Large-Eddy Smulaton of Channel Flow, Wth and Wthout Explct Flterng, Journal of Flud Mechancs, Vol. 495, pp Hughes, T., Mazze, L., Obera, A., and Wray, A. (2001) The Multscale Formulaton of Large Eddy Smulaton: Decay of Homogeneous Isotropc Turbulence, Physcs of Fluds, Vol. 13, pp Katopodes, F.V., Street, R.L., and Ferzger, J.H. (2000a) A Theory for the Subflter-Scale Model n Large-Eddy Smulaton, Techncal Report 2000-K1, Envronmental Flud Mechancs Laboratory, Stanford Unversty. Katopodes, F.V., Street, R.L., and Ferzger, J.H. (2000b) Subflter-Scale Scalar Transport for Large-Eddy Smulaton, 14th Symposum on Boundary Layers and Turbulence, Amercan Meteorologcal Socety, pp

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