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A new large-eddy simulation near-wall treatment

Two different types of instantaneous wall boundary conditions have been proposed for resolved large scale simulations that extend inside the viscous sublayer. These conditions transfer the physical no-slip and impermeability/permeability information,
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  A new large-eddy simulation near-wall treatment M. Iovieno,G. Passoni, andD. Tordella   Citation:Phys. Fluids 16 , 3935 (2004); doi: 10.1063/1.1783371   View online:   View Table of Contents:   Published by the American Institute of Physics.   Related Articles Inertial-range anisotropy in Rayleigh-Taylor turbulence   Phys. Fluids 24, 025101 (2012)   Modal versus nonmodal linear stability analysis of river dunes   Phys. Fluids 23, 104102 (2011)   The equivalence of the Lagrangian-averaged Navier-Stokes-α model and the rational large eddy simulationmodel in two dimensions   Phys. Fluids 23, 095105 (2011)    Amplification and nonlinear mechanisms in plane Couette flow   Phys. Fluids 23, 065108 (2011)    A shallow water model for magnetohydrodynamic flows with turbulent Hartmann layers   Phys. Fluids 23, 055108 (2011)   Additional information on Phys. Fluids Journal Homepage:   Journal Information:   Top downloads:   Information for Authors:   Downloaded 13 Mar 2012 to Redistribution subject to AIP license or copyright; see  A new large-eddy simulation near-wall treatment M. Iovieno  Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy G. Passoni  Dipartimento di Ingegneria Idraulica, Politecnico di Milano, Ambientale, Rilevamento Piazza Leonardo da Vinci 32, 20133 Milano, Italy D. Tordella a )  Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy ( Received 24 February 2004; accepted 4 June 2004; published online 5 October 2004 ) Two different types of instantaneous wall boundary conditions have been proposed for resolvedlarge scale simulations that extend inside the viscous sublayer. These conditions transfer thephysical no-slip and impermeability/permeability information, which can only be rigorously appliedto the unfiltered variables, to the filtered variables. The first condition is universal, while the secondone specifies the wall stress and relevant distribution and can be used to treat inverse flow problems.The filter scale close to the wall is a function which varies according to its position and thus theproblem of the noncommutation of the filter and differentiation operators arises. Used together withthe explicit noncommutation procedure by Iovieno and Tordella, these boundary conditionsconstitute a wall treatment which could improve the use of the large-eddy methodology in relationto aspects that are independent of the modeling of the subgrid scale motion. When applied in the testcase of the plane periodic channel, intentionally using the most crude subgrid scale model ( Smagorinsky, with no dynamic procedure or wall damping function ) to prove its efficacy, theproposed near-wall treatment yielded resolved large-eddy simulations which compare well withboth direct numerical simulations and with experimental data. The effects of the Reynolds numberon the structure of the flow are retained. Distributions of the noncommutation error on the turbulentsolution are also reported. © 2004 American Institute of Physics . [ DOI: 10.1063/1.1783371 ] I. INTRODUCTION The large-eddy simulation ( LES ) method is probably go-ing to be one of the most frequently used tools to predict thebehavior of turbulent flows for many different physical andengineering applications. Among these applications, wallflows constitute a separate class, due to the peculiarities of the near-wall dynamics that are related to important applica-tions in geophysics, hydrodynamics, and gasdynamics. Theturbulence near the wall is very unhomogeneous and not inequilibrium. The diffusive vorticity generation is coupledand is of the same order as the unsteadiness and nonlinearity.Such a complex situation is not easily synthesized in amodel, because, close to the wall, the categories on whichthe turbulence modeling of homogeneous or nearly homoge-neous flows relies are not valid, the conceptual separationbetween the large and small scales is not possible, and theasymptotics similarity is not observed in practical problems.It is crucial for physicists and engineers, who neverthelessmust produce approximate but reliable forecasts to improvethe use of the method as much as possible independently of the physical features of the subgrid scale model that isadopted. For this purpose it is important to consider the fol-lowing: ( 1 ) the transfer of the wall physical conditions,which can only be rigorously applied to the unfiltered vari-ables, to the filtered variables and ( 2 ) the noncommutationproperty loss between the filter and differentiation opera-tions, which affects the simulation of unhomogeneous fields,such as the wall flows, in which the filter scale varies greatlyaccording to the position ͓ ␦  = ␦  ͑  x i ͔͒ . 1–5 In this situation, thegoverning equations change structure, because a noncommu-tation term must be introduced in correspondence to eachspatial differential term. The change in the filtered governingequations introduces variations to their numerical solution.For high Reynolds number flows, the problem of theboundary conditions for the filtered field can be treated byadopting one of the classical approximated conditions thatrelies on the introduction of special wall models, which rep-resent the inner layer dynamics [ usually in a Reynolds-averaged sense, see the review by Piomelli and Balaras, Sec.2 ( Ref. 6 )] , and by putting the first grid point used by thelarge-eddy simulation inside the logarithmic layer ( see Ref.7, the wall-stress models by Schumann 8 and Piomelli et al. , 9 the two-layer models by Balaras et al. , 10 the detached eddysimulation approach by Baggett 11 and Nikitin et al. 12 ) . Thesemodels were conceived to avoid the prohibitively expensivecomputational cost of resolving the wall layer in high Rey-nolds number environmental and engineering applications.However, at a fundamental level, with regard to the LES a ) Author to whom correspondence should be addressed. Telephone: 0039011 564 6812; fax: 0039 011 564 6899; electronic mail:daniela.tordella@polito.itPHYSICS OF FLUIDS VOLUME 16, NUMBER 11 NOVEMBER 2004 1070-6631/2004/16 ( 11 )  /3935/10/$22.00 © 2004 American Institute of Physics3935 Downloaded 13 Mar 2012 to Redistribution subject to AIP license or copyright; see  methodology, and to resolve the near-wall dynamics, it isalso acceptable to place grid points inside the viscous sub-layer.The boundary conditions for the filtered variables shouldbe different from those that are canonical for the unfilteredvariables, i.e., the no-slip and impermeability conditions u i =0 at the wall. First, the filtering operation ( e.g., the volumeaverage ) is ill defined for grid points placed on the wallbecause, in this case, the filter width extends beyond the wallboundary ( i.e., outside the flow domain ) ; second, a filteringvolume in contact with the wall, but entirely merged withinthe domain, will give averaged velocities that are differentfrom zero and which are placed in the dynamical center of the average volume which will always be located at a finitedistance from the wall.On the other hand, the alternative option of the grid re-finement ( i.e., the filter width that goes to zero as the wall isapproached ) is not clearly defined. In this case in fact it is notpossible to automatically determine where the shift fromLES to DNS takes place. This shift would necessitate thechange of the evolution equations from the filtered NS ver-sion ( LES ) to the unfiltered NS ( DNS ) . Since this change isnot carried out, which would inevitably imply the introduc-tion of a domain decomposition, the simulation cannot beconsidered as being based on a consistent problem form.Furthermore, even in the hypothesis of having consistentlysplit the domain to carry out the hybrid LES-DNS, it wouldmean adopting a time step which must fit the DNS require-ments close to the wall. The temporal integration scale forthe DNS is faster than that required by the LES, but since itis not possible, advancing on time, to differentiate the tem-poral steps into different regions of the computational do-main, the DNS requirement would take precedence over theLES one, and this is not convenient.It is here proposed to shift the boundary conditions forthe filtered field onto a surface that lies on a first level of gridpoints and is parallel to the wall, at a distance of the sameorder as the viscous length. The transfer of the informationthat is relevant to the physical properties of the wall is ac-complished by considering a series expansion in ␦  for thefiltered variable, at the first layer of points. If associated to aTaylor expansion of the unfiltered variable at the wall, thisyields a first kind of condition that is universal in character.If the ␦  expansion is instead related to a Mac Laurin expan-sion of the unfiltered variable at the wall, a second kind of boundary condition is obtained which is suitable to impose aknown distribution of wall stresses, as normally asked in thecontext of inverse mathematical problems. The boundarycondition formulations are described in Sec. II. The related a priori tests, 13 which showed a correlation with DNS data 14,15 as high as 0.97 for a boundary shift of five wall units, aredescribed in Sec. II A.As previously explained, the other feature that has beenimplemented in the simulations is the noncommutationprocedure. 1 This is based on an approximation of the differ-ent noncommutation terms in the governing equations asfunctions of the ␦  gradient and of the ␦  derivatives of thefiltered variables. The anisotropic noncommutation approxi-mating terms, of the fourth order of accuracy in the filterscale, are obtained using series expansion in ␦  of approxima-tions based on finite differences and introducing two succes-sive levels of filtering. A brief outline of this procedure isgiven in Sec. III. The distribution of the noncommutationerrors on the Reynolds stresses is given in Sec. IV.The results that were obtained when using the presentwall treatment applied to the LES of the channel flow andobtained by utilizing the most crude SGS model ( Smagorin-sky, with no dynamic procedure or wall damping function ) are discussed in Sec. IV. The simulations compare well withthe direct numerical simulations 14,15 and with laboratoryobservations. 16–18 The simulations show the correct Rey-nolds number dependency. Given this, much greater progresscan be expected if the dynamic procedure 19,20 and modelswhich allow for significant nonlocal and nonequilibrium ef-fects are used. 21–24 The concluding remarks are given in Sec.V. II. NEAR-WALL TREATMENTA. Wall conditions for the filtered variables The shifting of the boundary conditions for the filteredfield on a surface that lies beside the wall at a distance y គ of the same order as the viscous length ᐉ ␶   ͓  y = O ͑ ᐉ ␶   ͒ = O ͑ ␦  min ͔͒ , see Fig. 1, is here proposed. The first level of thegrid points should in turn be positioned on the shifted bound-ary. This shift offers a twofold advantage—first, the depen-dent variables are correctly determined, as it is possible to seta local volume of integration which does not cut the physicalboundary; and second, since the shifted boundary is close tothe wall, it is possible to transfer the physical informationthat corresponds to the no-slip and impermeability/ permeability conditions to the shifted condition through aseries expansion. An expansion in series of  ␦  of the filteredvariable can in fact be associated, at the first layer of points,to a Taylor or a Mac Laurin expansion of the unfiltered vari-able at the wall, where the filter length reaches the minimumvalue normal to the wall ␦  min . While using this boundaryformulation, one is aiming at simulating the inner viscouslayer in the region y + Ͻ 50, which requires ␦  min =  y គ ϳ 1–5. FIG. 1. A schematic view of the shifted boundary conditions, the filter, andthe grid. 3936 Phys. Fluids, Vol. 16, No. 11, November 2004 Iovieno, Passoni, and Tordella Downloaded 13 Mar 2012 to Redistribution subject to AIP license or copyright; see  Let us consider the class of integration volumes V  ␦  = ͭ ␰  ෈ R 3 : Ͱ ͩ ␩  1 ␦  1 , ␩  2 ␦  2 , ␩  3 ␦  3 ͪ Ͱ Ͻ 1 ͮ , ͑ 1 ͒ where ␦  ͑ x ͒ = ( ␦  1 ͑ x ͒ , ␦  2 ͑ x ͒ , ␦  3 ͑ x ͒ ) and the transformation ␩   j = ␦   j ␰   j [ with det ͑ ␩  i  /  ␰  k  ͒ = ␦  1 ␦  2 ␦  3 and where no summationis implied ] has been introduced. Let us consider the averageoperation for the variable f  ͑ x ͒ =  f  ͑  x  j + ␦   j ␰   j ͒ : ͗  f  ͘ ␦  =1 V  ␦  ͵ V  ␦   f  ͑ x + ␩  ͒ d  ␩  =1 V  1 ͵ V  1  f  ͑  x  j + ␦   j ␰   j ͒ d  ␰  , ͑ 2 ͒ where 1 = ͑ 1,1,1 ͒ and V  1 = V  ␦   /  ␦  1 ␦  2 ␦  3 .For the sake of simplicity, let us now consider the caseof a flat wall flow. Here the filter can be opportunely repre-sented by the widely used notation ␦  ͑ x ͒ = ( ⌬  x , ␸  ͑  y ͒ ⌬  y , ⌬  z ) with constants ⌬  x , ⌬  y , and ⌬  z and where y is the coordinatenormal to the wall. The ␦  expansion for a general variable f  ,after setting ␦  ͑  y គ ͒Х ␦  min = ␸  min ⌬ , yields ͗  f  ͑͘  y គ ͒ = f  ͑  y គ ͒ + L ͓  f  ͔ ␦  min2 + O ͑ ␦  min4 ͒ , ͑ 3 ͒ where  L ͓ · ͔ = 12 a ˜ 2 , ˜ =  y 2 2 + ͩ ⌬  x ␦  min ͪ 2  x 2 2 + ͩ ⌬  z ␦  min ͪ 2  z 2 2 , ͑ 4 ͒ a =1 V  1 ͵ V  1 ␰  i 2 d  ␰  , ͑ 5 ͒ and where it should be recalled that, due to ( 1 ) and ( 2 ) , thecoefficients of the odd powers of  ␦  min are zero. By applyingthe operator L to ( 3 ) one obtains  L ͓͗  f  ͔͘ = L ͓  f  ͔ + L † ␦  min2  L ͓  f  ͔ ‡ + O ͑ ␦  min4 ͒ = L ͓  f  ͔ + O ͑ ␦  min2 ͒ . ͑ 6 ͒ As a consequence, ( 3 ) can be written as  f  ͑  y គ ͒ = ͗  f  ͑͘  y គ ͒ − ␦  min2  L ͓͗  f  ͔͘ + O ͑ ␦  min4 ͒ . ͑ 7 ͒ In turn, to transfer the no-slip and impermeability informa-tion, which applies at y =0, let us consider the Taylor expan-sion along the normal to the wall,  f  ͑ 0 ͒ = f  ͑  y គ ͒ − y គ y  f  ͑  y គ ͒ +  y គ 2 2 y  f  ͑  y គ ͒ + O ͑  y គ 3 ͒ . ͑ 8 ͒ Since relation ( 6 ) can be generalized as d  m dx m  f  ͑  x  j ͒ = d  m dx m ͗  f  ͑͘  x  j ͒ + O ͑ ␦  2 ͒ , m = 1,2, … , ͑ 9 ͒ expansion ( 8 ) can be written as  f  ͑  y គ ͒ = f  ͑ 0 ͒ + y គ y ͗  f  ͘ −  y គ 2 2 y 2 ͗  f  ͘ + O ͑ ␦  min3 ͒ . ͑ 10 ͒ By equating ( 7 ) and ( 10 ) , while recalling definition ( 4 ) , andtruncating the third-order terms, a new boundary condition isobtained at y =  y គ , ͗  f  ͘ = f  ͑ 0 ͒ + y គ y ͗  f  ͘ +12 a ˜ 2 ͓͗  f  ͔͘ ␦  min2 −  y គ 2 2 y 2 ͗  f  ͘ , ͑ 11 ͒ where the filtered variable at each instant depends explicitlyon the position y គ , the values of its first and second deriva-tives, and the wall value of the unfiltered variable, whichintroduces the physical information. This condition—whichin the following is called condition I—is an instantaneouscondition and it is universal because it can be applied to anykind of wall boundary. One should note that, according to thetheory established by Kreiss 25 for Dirichlet differential prob-lems discretized with a finite-difference scheme with order O ͑ ␦  p ͒ at inner points and O ͑ ␦  p −1 ͒ at points close to theboundary, the error of the discrete solution is O ͑ ␦  p ͒ through-out. Therefore, with such a boundary condition formulationand truncating the terms of order O ͑ ␦  3 ͒ , the fourth order of accuracy reached for the approximation of the noncommuta-tion term proposed by Iovieno and Tordella 1 can be expectedto be preserved. In the present simulation, the y derivativesof the filtered variables at y =  y គ have been calculated usingone-sided discrete operators.Another similar type of boundary condition can be writ-ten by equating ( 7 ) to the Mac Laurin series f  ͑  y គ ͒ =  f  ͑ 0 ͒ +  y គ y  f  ͉  y =0 + ͑  y គ 2  /2 ͒  y 2  f  ͉  y =0 + O ͑  y គ 3 ͒ , which gives the condition ͗  f  ͘ = f  ͑ 0 ͒ + y គ y ͉  f  ͉  y =0 +  y គ 2 2 y 2 ͉  f  ͉  y =0 + a 2 ␦  min2 ˜ 2 ͓͗  f  ͔͘ . ͑ 12 ͒ In the case of a steady ( in the mean ) turbulent flow it ispossible, with this formulation, to transfer the informationrelevant to the time average of the wall shear stress distribu-tion and its derivative to the filtered field. Condition ( 12 ) —which in the following is called condition II—is an instanta-neous condition. However, it is physically reasonable toinsert the time averaged values of the wall stress only in thecase of flows which are steady in the mean, as is the case of the example that we have considered in this paper. However,this is not the only possibility. In fact, if detailed informationon the temporal and spatial variation of the wall stress areavailable, it would in fact be better to insert them into infieldcondition II. The innovative and srcinal character of condi-tion II is that it allows the flow to be fed with different wallinformation, which, apart from the pure no-slip condition,also includes information on the time and space fluctuations,or possible evolutions of the wall stress along the wall. Allthis can be gathered in one single condition. This is feasiblebecause, having placed the condition relatively close to thewall—at a distance of almost one viscous length—one canuse a Mac Laurin series expansion to transfer the physicalinformation at the wall to the infield condition. In otherwords, a convenient situation is obtained: instead of usingthe physical boundary—the wall—where only one boundarycondition ( bc ) can be placed for each variable ( on the vari-able itself or on one of its derivatives in the direction normalto the wall ) , one can give a plurality of information concern-ing the values of the variables and of their relevant wallderivatives to the field, with just one condition and withoutoverconstraining the flow. Phys. Fluids, Vol. 16, No. 11, November 2004 A new large-eddy simulation near-wall treatment 3937 Downloaded 13 Mar 2012 to Redistribution subject to AIP license or copyright; see  A solution accuracy of the fourth order could also beexpected when using this boundary condition ( see the previ-ous comments ) . It must be noticed that, with this kind of formulation, the shear stress distribution along the wall canbe imposed to the field, as can the related characteristics suchas the intensity of the wall roughness. In this case, thisboundary condition can be applied to inverse mathematicalproblems. B. A priori  test on the approximate boundaryconditions The correlation between the filtered values that are ob-tained by the present shifted boundary conditions and thefiltered values that are obtained by a direct numerical simu-lation can be defined, as a function of the distance of theshifted boundary from the wall, as C  = ͑͗ u i ͘  DNS  − ͗ u i ͘  DNS  ͒͑͗ u i ͘ − ͗ u i ͒͘  /  ͱ  var ͑͗ u i ͘  DNS  ͒ var ͑͗ u i ͒͘ , ͑ 13 ͒ where ͗ u i ͘  DNS  are the filtered data from the direct simulationdata base ( Passoni et al. 15 ) , ͗ u i ͘ have been computed from ( 11 ) and ( 12 ) by introducing the filtered direct simulationdata into the right-hand side, the overbar means the averageover the surface parallel to the wall, and var is the variance.For Re ␶   =180 ( the Reynolds number for which a wide set of instantaneous fields was available ) Fig. 2 shows that bothboundary conditions I and II yield a correlation C  which isover 0.97 for the three velocity components up to a distanceof five wall units and goes down to 0.8, 0.88, 0.7 for ͗ u ͘ , ͗ v ͘ , ͗ w ͘ , respectively, at seven wall units. C. Noncommutation treatment The noncommutation procedure and the new wall condi-tion models together constitute the new treatment for wallturbulence that is here proposed. The main applications thatrequire a highly variable filtering in LES are wall flows.They must be represented through ( a ) accessory conditionsthat are consistent with the LES methodology, on the onehand, and ( b ) , on the other hand, a possibly explicit noncom-mutation procedure. As explained in the Introduction, apartfrom the matter relevant to the quality of either dynamical ornot subgrid model, the no-slip condition associated to the useof a filter width which goes to zero as the wall is approachedis not a fully consistent treatment for large-eddy simulationsof near-wall turbulence.A brief description of the noncommutation procedure isgiven in the following. When ␦  , the linear scale of filtering,is not uniform in the flow domain, the averaging and differ-entiation operations no longer commute. This leads to inho-mogeneous terms in the motion equations that act as sourceterms whose intensity depends on the gradient of the filterscale ␦  and which, if neglected, induce a systematic errorthroughout the solution. One kind of noncommutation termfor each differential term is present in the governing equa-tions. By introducing the subgrid turbulent stresses  R ij ͑ ␦  ͒ = ͗ u i ͘ ␦  ͗ u  j ͘ ␦  − ͗ u i u  j ͘ ␦  and the noncommutation terms C  i Ј , C  ii Љ , for the first and second derivatives, as discussed byIovieno and Tordella 1 ( see there, Sec. II for the isotropicfilter configuration and the Appendix for the general aniso-tropic and the wall-bounded flow configurations ) the aver-aged governing equations can be written as i ͗ u i ͘ ␦  = − C  i Ј ͑͗ u i ͘ ␦  ͒ , ͑ 14 ͒ t  ͗ u i ͘ ␦  + j ͑͗ u i ͘ ␦  ͗ u  j ͘ ␦  ͒ + i ͗  p ͘ ␦  − ␯   jj 2 ͗ u i ͘ ␦  − j  R ij ͑ ␦  ͒ = − C   j Ј ͑͗ u i ͘ ␦  ͗ u  j ͘ ␦  ͒ − C  i Ј ͑͗  p ͘ ␦  ͒ + ␯  C   jj Љ ͑͗ u i ͘ ␦  ͒ + C   j Ј ͑  R ij ͑ ␦  ͒ ͒ . ͑ 15 ͒ The present procedure approximates the noncommutationterms on the right-hand side of  ( 14 ) and ( 15 ) by means of series expansion in ␦  of finite difference approximations of the space derivatives and by introducing a successive level of filtering. An accuracy of the fourth order is reached withrespect to ␦  .Let us recall that the anisotropic noncommutation termof the first derivative C  i Ј , defined by FIG. 2. An a priori test on the approximate boundary conditions: correlationlevel as a function of the distance y min =  y គ of the shifted boundary from thewall. Top panel condition I, Eq. ( 11 ) ; bottom panel condition II, Eq. ( 12 ) . 3938 Phys. Fluids, Vol. 16, No. 11, November 2004 Iovieno, Passoni, and Tordella Downloaded 13 Mar 2012 to Redistribution subject to AIP license or copyright; see
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