A Novel Explicit Filter for the Approximate Deconvolution in Large-Eddy Simulation on General Unstructured Grids: A posteriori tests on highly stretched grids

A N ovel Explic it F ilter for the Approximate Deconvo lution in Lar ge - E ddy S imulatio n on General Unstruc tured Grids : A posteriori tests on highly stretched gri ds Mohammad Bagher Molaei 1 , Ehsan Amani 1, * , and Morteza Ghorbani 2 1 Department of Mecha nical Engi neering, Amirkabi r University of Tech nology (Tehra n Polytechnic), Ira n 2 Sabanci Uni versity Nanote chnology Rese arch and Applic ation Center , 34956 Tuzla, Is tanbul, Turk ey Abstract Explicit filte rs play a pivotal role in the scale separation an d numerical stability of advanced Large Eddy Simulation (LES) closures, such as dynamic eddy- viscosity or Approximate Deconvolution (AD) methods . In the present study , it is demonstra ted that the performance of commonly used explicit filte rs applicable to general unstructured grids highly depends on the grid configuration, specifically the cell asp ect ratio, which can result in poor filter spectral pr operties, ultimately leading to large error s and even solution divergence. This study introduces a novel , efficient explicit fil ter for general unstructured grids , addressing th is shortcoming through a combination of a face - averaging technique and recursive filtering. The filter para meters are then determined through a constrain ed multi - obje ctive optimizat ion , ensuring desirable spectr al properties, including high - wavenumber attenuation, filte r - width precision, filter st ability and positivity , and minimized disp ersion and commutation errors. The AD - LES of turbulent channel flow benchmarks using the new filter demonstrate a noticeable imp rovement in turbulent flow prediction s on highly stretched boundary - layer - type grids , particula rly in reducing the log - layer mean veloci ty profile mismatch, com pared to simulations using conventional fil ters. Th e analyses show that this enhancement is mainly attributed to the suf ficient level of attenuati on near the Nyquist wavenumber achie ved by the new fi lter in all spatial d irections across various grid configurations , among others . The new filter wa s also successfully tested on unstructured prism grids for the 3D Taylor-Green vortex benchmark. Keywords: Lar ge- Eddy Simul ation (LES); Ap proximate D econvolution (A D); Explic it Filter ; Unstructure d grid * Corresponding author . Address: Mechanical Engineering Dept., Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenu e, Tehran, P .O.Box: 15875- 4413, Iran. Tel: +98 21 64543404. Email: eamani@aut.ac.ir 2 1. Introduction Large Eddy Simulation (LES) has emerged as a pivotal tool for analy zing turbulent flow s . It is based on t he decomposition of flow properties into two contrib utions : a larg e - scale or resol ved component, and a small - scale or S ub-G rid - Scale ( SGS) on e ( Pope 2000 ) . This separation is accomplished through the application of a low -p ass spat ial filter. In the conventional LES framework, the imp licit or grid filter ing process is int rinsically e mbedded within the numerical discretizat ion of the dif ferential eq uations. H owever, a si gnificant limit ation of this i mplicit filtering app roach lies in its inherent inco nsistencies, p articular ly its inabili ty to achieve a well - defined 3D f ilterin g effect ( Lund 2003 ) . Moreover, the absence of control over the frequency content of the solution c an increase aliasing errors , leading to numer ical instabi lit ies , and introduce sensitivity to grid resolution ( Vasilyev et al. 1998, Chow & M oin 2003 , Ki m et al. 2021) . Additionally , the numerica l discretizat ion errors asso ciated with spatial deriva tives play a significant r ole in the overall LES error. An analysis of modi fied wavenumbe r plots for discretization scheme s ( Stolz et al. 2002 ) reveals that these errors are particularly pronounced for wavenumbers approaching the Nyquist frequency. Consequently, minimizing or eliminating the influence of such wav enumbers o n the solution can enhance the accu racy and reliability of LES results. A possible approach to mitigate t he grid dependency and discretization errors of the scal es close to the Nyquist is the application of an explicit low - pass fil ter with a cut - off wavenumber substantially lower than the Nyquist wavenumber ( Lund & K altenbach 1995 , Najjar & Tafti 1996 , Bose 2012 , Kim et al. 2021, Amani et al. 2024) . By applying an explicit filt er, an addition al scale sep aration is achieved, creating a S ub -F ilter S cale (SFS) alongside the conventional SGS . While the SGS remains inherently unresolved and must be approximated through modeling, the SFS can, in principle, be recon structed theore tically ( Gullbrand & Chow 2003 ) . On the other hand, t he low - pass filter p lays a pivotal ro le as the tes t filter in dy namic SGS mo de ls ( Sarwar et al. 2017) , prompting e xtensive research efforts to op timize its desig n and implement ation (Germano 1986b, c, Najjar & Tafti 1 996 , Sagaut & Grohens 1999, Pruett & Adams 2000, Berland et al. 2011) . A prominent LES strategy that leverages this idea is the Approximate D econvolution (AD) method ( Stolz & Adams 1999 ) , involving a two - stage process: the soft dec onvolution, which deals w ith the SFS reconstruction (up to the Nyquist wavenumber) from the solut ion using an (approximate) inverse explicit filter, and the hard deconvolution — also referred to as (secondary) 3 regularizatio n — which addres ses SGS modeling. Several characteristics are known to be desirable for explicit filters used for AD - LES , including no altera tion at low to moderate wave numbers, a large attenu ation near the Nyquist, a sharp roll - off, a low commut ation error, a small dispersion error, no amplification at any wavenum ber (filter stabili ty) , no transfer function overshoot, the positivity , the smoothness, a st rong anti - aliasing, etc. ( Vasily ev et al. 1998, Sagaut 2006 , Layton & Rebholz 2012 , Najafiyazdi et al. 2023) . Originally, Stolz and Adams (1999) use d the im plicit Padé filters for AD - LES . In a pioneering work, Vasilyev et al. (1998) introduc ed criteria in terms of filter moments to achieve a desired order of the comm utation error. They proposed discrete filters in compu tational space , imposing thes e criteria along with additional constraints to adjust a prespecified filter wi dth and ensure perfe ct attenuation at the Nyquis t. S tolz et al. (200 1) extended this method by defining discrete f ilters in ph ysical space . They enforced a higher order of commutatio n error at th e boundaries, compared to int ernal nodes. They also used an additional constraint to minimize the dispers ion error (the im aginary part o f the filter transfe r function). They showed that d efining filter mo ment criteri a in physical sp ace is superior to the original f ormulation in computational space ( Vasilyev et al. 1998) , since the forme r is computation ally more ef ficient and conditions for a small altera tion at low - wav enumbers are au tomatically met for the f ilters with an explicit rule . In addition, they designed new ( implicit - form ) Padé filters to meet al l these criteria. All these f ilters w ere restricted to structured grid s. Early attempts to design explic it filters for unstru ctured grids da te back to the w ork by Marsden et al. (2002) and Haselbacher and Vasilyev (2003) . They developed 2 nd - order filt er s fo r unstructured grids through a local neighbor nodes selection procedure with a 3 - node triangular stencil in 2 D or 4 -n ode tetrahedral stencil in 3D . The first group ( Marsde n et al. 2002) employed linear combinations of polynomial interpolation functions , and the second group ( Haselbacher & Vasilyev 2003 ) adopted the least - square s gradien t reconstruct ion method. However, these filter’s properties we re sensitive to grid node distribution. In addition, constructin g geometric s implices — especially in 3D or near boundaries —add ed the c omplexity and declined the stability (especially in the case of skewed or stre tched elements) in the se methods. The differen tial filter ing , originally proposed by Germano ( 1986a, c) , is an alternat ive solution for unstructured grids. San et al. ( 2015) compared the performance of the Padé , box, and elliptic differentia l filters for AD - LES of Taylor - Green vortex and pointed out that hyper - differentia l filters perform better than the others. However, in their subs equent studies ( San 2 016 , Maulik & 4 San 2018 ) on filter types, including Padé, hyper - differentia l, and discre te binomial, for decayin g Burgers turb ulence, they concluded th at the Padé filter with  = 0.3 outperforms the others. Chang et al. ( 2022) s tudied some invertible filter types, including Gaussian, dif ferential, hyper - differential, Chebyshev, etc., for the approximate and direct deconvolution methods in LES of isotropic tur bulence. Th ey reported that d ifferentia l filters need a la rger filter - to - grid size ratio compared to the others to give rise to accu rate re construction s. D ifferential filter s do not possess some charact eristics desired for AD - LES , e.g., they lose commutativi ty, do not have a large attenuation at the Nyquist, and are rather dissipa tive for low to moderate wave numbers. Recently, Najafi - Yazdi et al. ( 2015, 2023) tackled the latter t wo issues by design ing an improved differenti al filter in the co ntext of the finite element discre tization wit h several specific element topologies and shape functions. The i ncorporation of th ese filters in to F inite V olume Method (FVM) solvers , or th eir applic ation with gen eral grid to pologies or discretization schemes, needs further investigation. In addit ion , these filters introduce ch allenges at non - periodic boundaries, such as walls, where the velocity gradient must be provided or numerically reconstru cted ( Bose 2012 ) , which produces an additional uncertainty. Therefore, despite the w ide usage of differen tial filters ( Vr eman 2004 , Nouri et al. 2011, Aljure et al. 2014 , López Casta ño et al. 2019, Saee dipour et al. 2019, Schneiderbauer & Saeedipour 2019 , Rauchenzauner 2022 , Rauchenzauner & Schneiderbauer 2022 ) , the lack of commutativity, a high computational cost, a poor parallelizability, and the introduction of errors d ue to uncertainties in the numerical tre atment of the diffe rential filte r at wall boundaries are the general drawb acks of these filters, yet to be addressed. Other impor tant studies on th e design of explici t filters includ e the work by Kim et al. (2021) , who improved the parallel effic iency via a recursive filt ering approach , repla cing wide - stencil filters with repeated narro w - stencil ones . This redu ced memory usage and inter - proces s communication but increased arithmetic operations by ~25% and remained limited to structured grids. T he divergence - preserving property of filters was highlighted by Agdestein and Sanderse (2025) . They proposed a filter on unstructured grids using face averaging , rather than volume averaging , along with grid cell agglo meration . Based on our literature review, the des ign and implemen tation of ef ficient explicit filters specifically tailored for FV M - based solvers on general unstructured grids remain underexplored. Here, we aim to design a new explicit filter for AD - LES on unstructured grids ( or non - uniform 5 structured grids with high aspect ratio s) , partially ad dressing the de ficiencies o f existing fil ters . Most notably, while recursive filters have shown promise in the structured grid context ( Kim et al. 2021) , their adaptation to unstructured grids and their optimization for accuracy and stability remain unexplored. This study is conducted towar d s filling these gaps by proposing a novel recursive filt er on uns tructured grids leveraging a multi - objectiv e optimization framework to fine - tune filter coefficients fo r desirable properties. The remainder of this p aper is structured as follow s: Sec tion 2 provides the mathematical modeling of AD - LES , benc hmarks used for a posteriori t ests, and numerical methods . Section 3 discusses explicit fil ters for AD - LES , outlining the filter design c riteria, analyz ing the limita tions of existing filtering approaches, and int roducing a novel recursive f ilter that addresses these challenges while meeting the desired properties. Section 4 features the results of the a post eriori analyses tha t rigorously evaluate the n ew filter performance , in comparison to the widely use d filters for general unstructured grids. Finally, s ec tion 5 summarizes the key findings and overarching conclusions of this st udy. 2. AD - LES modelling 2.1. Mathematical modelling The Navier -Stokes (NS) equations governing incompressible flows of a Newtonian fluid are expressed as : (1)     +        +        2          +  (   ) = 0, (2)     = 0, where   and   denote th e time (  /  ) and spatial (  /    ) derivative s , respectively . The se equations include the velocity vector ,   , the kinematic pressure ,  , the kinematic viscos ity ,  , and the strain - rate tensor ,   , defined as: (3)   = 1 2     +      . The term "kinema tic" refers to th e division of th e correspondin g variable by the constant fluid density,  . For brevity, the te rm "kinematic" is omitted h ereafter a nd assumed imp licit for al l pressures, viscosities, an d stresses. The filtered NS equations solved in an AD-LES approach can be written as: 6 (4)       +            =      +    2   󰆻         , (5)       = 0 , where (. )  denotes the grid filter, (. )     or   (. ) the explicit filter where  represents the explicit filter kerne l, and   is the Sub - Filter - Scale (SF S) stress tensor defined by: (6)                        =   +   . To solve Eq s. (4) and (5) for     and    , a closure model is requ ired for the unc losed SFS stress ,   , (or     ). This quantity is decomposed into two part s, the deconvolved SFS stress,   , and the modeled SFS stress ,   . A widely - used decomposition of   can be written as ( Carati et al. 2001) : (7)   =                     , (8)   =                       =   , where   is the SGS stress tensor def ined by: (9)   =             . Here, t he deconvolved SFS stress is reconstructed by the Scale - Similarity ADM ( SSADM ) soft deconvolution (Stolz & Adams 1999) : (10)    =                          , where the superscript  denotes the modeled counterpart of a quantity, and    is t he (deconvolved) approximation to    . We use the Van Citter t iterative alg orithm ( Van Citte rt 1931 ) with five iterations for the approximate deco nvolution operation . The modeled SF S stress,   , encapsulates the SGS effect and must be modeled (    ) using an SGS closure or (secondary) regularization. For this purpose, the mixed Alternative Linear Dynamic Mo del with Equilibr ium assumpti on (ALDME) model proposed by Aman i et al. (2024) is employed, where (11)    ,  =    ,  =      ,  , (12)    ,  = 2           = 2      , where superscript  refer s to the deviatoric part of a tensor ,   =  2         /  ,   the grid filter wid th ,   the SGS eddy-viscosity, and   the dynamic model coefficient computed by: ( 13 )   =          , (14)   =                 , 7 (15)   =  8           , and (. )  is a test box fil ter of width   = 2  󰆻 . The flow statistics are computed using the ensemble (Reynolds) averaging. Key quant ities include the mean vel ocity     and the Reynolds stress,    󰆒   󰆒  , where the fluctuat ing velocity is   󰆒 =        and   denotes the ensemble averaging operator. In the context of AD - LES, these quantities are approximated under the assumption          (Pope 2000) , as follows:            ;        󰆻    , (16)    󰆒   󰆒    (            )              +     +      +               , (17) where   represents the SGS Turbulent Kinetic Energy (TKE). For   , the following estimation can be emplo yed (Sullivan et al. 2003) :                  , (18) where   = 1 represents the model cons tant. 2.2. The a posteriori benchmarks 2.2.1. Turbulent channel flow The fully - develo ped turbulent flo w of a Newton ian fluid between tw o parallel infini te plates, driven by a constant mean pressu re gradient — commonly termed "channel flow" — is a benchmark problem frequently employed for both a posteriori and a priori analyses. The flow dynamics are characterized by a single dimensionless parameter, the friction Reynolds number, defined as    =    /  , where  is the half - channel height,   =    /  the friction velocity, and   the wall shear stress. In thi s study, the Direct Numeric al Simulation (DNS) dataset b y Moser et al. (1999) is used as the refere nce solution . The computat ional domain, illustrated in fig ure 1 a, is configured with dimensions of (  × 2  ×  ) = (2  × 2 ×  ) . The results are presented as a function of the normalized wall dis tance,   =   /  , where  represents the distance from the wall. 8 (a) (b) Figure 1: a) A schematic geometry of the channel flow and Q- criterion iso - surfaces predict ed by the present AD - LES, and b) a com putational grid used for simulations. (   = 395 ). 2.2.2. Taylor- Green vortex The 3D Taylor - Green V ortex (TGV) is a canonical benchmark, extensiv ely used to investigate vortex dynamics. Th is prob lem involves the decay o f a 3D field of vor tices within a cubic do main, subject to periodic boundary conditions on all faces. The computational domain, illustra ted in figure 2 , is a cube with a side length of 2  , where  serves as the re ference leng th. The ini tial flow field at  = 0 comprises a d istribution of distinct large - scale vortical s tructures defined by the following velocity field (DeBonis 2013) :  =   sin 󰇡   󰇢 cos 󰇡   󰇢 cos 󰇡   󰇢 ,  =   cos 󰇡   󰇢 sin 󰇡   󰇢 cos 󰇡   󰇢 ,  = 0, (19) where   denotes the initial velocity amplitude. The corresponding initial pressure field is analytically obtained to satisfy the Po isson’s equa tion for incomp ressible flow s as :  =   +      16  cos  2    + cos  2     cos  2    + 2  (20) where   and   are the reference pressure and density, respe ctively. The flow dynamics is characterized by the Reynolds number,  =      /  . The reference values are chosen according to ( DeBonis 2013 ) as:  = 0. 001524  ,   = 34 . 396  /  ,   = 0 Pa ,   = 1.0   , and  = 3. 2762 × 10      , leading to  = 1600 . A r eference DNS solution using a high -order 9 (4th- order in space and 3rd- ord er in time) finite - difference discretization on uniform struct ured grids of 512 × 512 × 512 cubic cells is available (DeBonis 2013) . Figure 2 : The computational domain and unstructured grid composed of prism cells , used for the current LES of TGV benchmark. 2.3. Numerical methods In this study, the cell - centered collo cated FVM imp lemented for gen eral unstruc tured grids in th e OpenFOAM ESI CFD pack age ( www.openfoam.com ), version 1912, was enhanced to incorporate ADM - type closures and the new filtering operation. For the present LES of channel flow, th e pressure - velocity coupling is re solved using th e Pressure Imp licit with S plitting of Op erators (PISO) algorit hm ( Issa 1986 ) e mpl oying two pressure - correcti on loops. Temp oral discretiz ation is achieved using the implicit second - order b ackward scheme ( Greenshields 2015 , Moukalled et al. 2016) . Gradients are computed via the G reen - Gauss cell - based method, utilizing the "Gauss linear" scheme ( Greenshields 2015 ) . The advection term in the momentum equation is discretized using the second - order pure - centered "Gau ss linear" sch eme, while variab le interpola tion to cell faces is performed us ing the "line ar" scheme. F or the pres sure equation, the resulting syste m of linear algebraic equations is solved using the Preconditioned Conjugate Gradient (PCG) method, with the Diagona l Incomplete - Cholesky (DIC) preconditioner combined with Gauss - Seidel smoothing ( Saad 2003 ) . The momentum equation is solved using the Stabilized Preconditioned Bi -Conjugate Gra dient (PBiCG Stab) solver, pre condition ed with the D iagonal Incom plete - LU (D ILU) method X Y Z 10 ( Van der Vorst 1992 , Barrett et al. 1 994 , Saad 2003 ) . A normalized residual to lerance of 10  is set as the convergence criterion for all vari ables at each time step. To ensure numerical stab ility and accuracy, the t ime step is dyna mically adju sted to maintain a maximum Coura nt - Friedric hs - Lewy (CFL) number of 0.5. The computational domain is depicted in figure 1 a . Periodic boundary conditions are applied to the streamw ise and spanwise boundaries for a ll flow variables , with the mean pre ssure gradien t enforced through an additional source term in the momentum equation. At the lower and upper walls, the no - slip cond ition is impo sed on the velo city field, w hile a zero - gradient condition is applied to th e pressure. T he grid is uniformly distributed in the streamw ise and spanwis e directions, with wall - normal refinement achieved using a hyperbolic tangent function to clu ster points near t he walls. Details o f the LES grid , along with the re ference DNS grid ( Moser et al. 1999) , are provided in table 1 . As can be seen in this table, the grid has a high aspect ratio w hich is typical of boundary layer grids used for LES a nd DNS. The parameters (   )  and (   )  in table 1 refer to the grid cell size (in the y directi on) on the wall and at the c hannel center plane, respectively. The computational grid topology is illustrated in figure 1b. Table 1 : A compariso n of the presen t LES and referen ce DNS ( Moser et al. 1999 ) grids .   ×   ×     /     [ (   )  , (   )  ]   Solution  ×   ×  =  ,  ,  1 6.5 [0.03, 6.5] 10 DNS (    = 395 )  ×  ×  =  ,  3.90 25 . 85 [0.82, 25 ] 38 . 78 LES (    = 395 )  ×  ×  =  ,  ,  1 4.8 [0.044, 7 .2] 9.7 DNS (    = 590 )  ×  ×  =  ,  4.72 25 [0.8, 35 .4 ] 40 LES (    = 590 ) To accelerate the simu lations, the Wall - Adap ting Local Eddy - viscosity (WALE) mod el ( Nicoud & Ducros 1999 ) is utilized for generating an initial solution for the AD - LES, due to its robustness and simplicity. The velocity field initialization procedure and the solution steps to achieve a WALE model solution have been deta iled elsewhere ( Amani et al. 2023 , Taghvaei & Amani 2023) . Starting from the WALE solution, the simulations continue for 30  , whe re  is the Flow -Through- Time (  =  /   ) and   is the channel bulk velocity, marking the attainment o f a statistic ally statio nary condition . Subsequent ly, the simu lations cont inu e to collect the statist ics by the e n sembl e averaging,   , taken here as a spat ial averaging in t he two homogeneous directions (  and  ) and a time - averaging for 70  . Concerni ng the validation of the unstructured - grid finite volume solver, i.e., OpenFOAM, the present solver has been 11 extensiv ely validated fo r LES of turb ulent flows in references ( Tofigh ian e t al. 2019, Amani et al. 2023, Taghvaei & Amani 2023) . Diverging solutions were observed for conventional pure Dynamic Eddy - Viscosity (DEV) and mixed AD - DEV models, see also ( A mani et al. 2024) . As a potential solution, the clipping method based on the realizability conditions proposed by Mokhtarpoor and Heinz ( 2017) was implemented, but it was not sufficient to resolve the instab ility issue for many cases. S ubsequently, the Positive Total Viscosity (PTV) approach was employed as   +   0 or   /    1 , which successfully addressed t he stability concerns. For the TGV benchmark, the numerical setup is modified to address the specific stability requirements of the transitional vortex breakdown. The pressure - velocity coupling within the PISO algorithm is augmented to include two non- orthogonal corrector steps to enhance accuracy on unstructured computationa l grid s . Regarding s patial discreti zation, the gradien t of the velocity field is computed using a cell - limited Gauss linear scheme ( Guo & Wang 2025 ) with a limiting coefficient o f 1.0 to mit igate numer ical oscillat ions. The adv ection term in the momentu m equation em ploys the Linear - Upwind Stab ilized Transpo rt (LUST) sch eme ( Cao & Tamura 2016 ) — a fixed blend of linear and l inear - upwind schemes — replacing the pure - centered approach used f or the channel flow benchmark to optimally balance dissipation and dispersion errors. Furthermore, to capture the rapid tempora l evolution of the small - sca le structures, the time st ep is dynamically restricted to main tain a maximum CF L number of 0.1. T he e nsemble averaging,   , for the TGV problem is taken as the volume - averaging within the whole computational domain ( DeBonis 2013) . 3. Explicit filt ers for AD - LES 3.1. The filter design criteria A 3D discrete explicit filter can be expressed as a weighted sum of values at cell centers, as fol lows (Sagaut & Grohens 1999) : (21) (    )  =    =    ,        , where    represents the fi ltered variable at ce ll  , the summation is over a ll grid cells, and   ,  denotes the weighting (or filter) coefficien t associated with cell s  and  . The discrete filter's transfer function in wavenumber space is defined by ( Sagaut & Grohens 1999 ) : 12 (22)   (  ) =    ,    .(     )     , where   is the position v ector ( of the center) of ce ll  and  is the wavenumber vector. The bold symbols indicate vector quantities , ( . )  the complex numbers , and  =   1 . The dimensionless wavenumber can be defined by  =     ,     ,      ( Sengupta & Bhumkar 2010 ) , where   is the characteristic g rid spacing in the  th - directi on , thus: (23)  = 󰇧     ,     ,     󰇨 . For AD filte r s , several desirable properties have been extens ively discussed in the literature ( Vasilyev e t al. 1998, Sagaut 2006 , Layton & Rebholz 2012 , Najafiyazdi et al. 2023) and are outlined as follows: 1. Normali zation: The filter should not alte r a uniform field , (24)   (  = 0 )  = 1     ,      = 1. 2. High -wavenumber a ttenua tion: Vanishing t he filter transfer funct ion at the Nyquist wavenumber ( |  | =  ), (25)   ( |  | =  )  = 0. 3. Preadjusted fil ter width : The filte r transfer funct ion must atta in a value of 0.5 at the specified cutoff frequency |  | =   , (26)   ( |  | =   )  = 0.5. 4. Commutativ ity: For non- uniform co mputational grids, it is es sential to pres erve the commutativi ty property, the communicat ion of the fi lter and diffe rence opera tors, to a reasonable degree. To a chieve  (   ) accuracy, the filter's momen ts (up to order   1 ) in the physica l space must be zero. For a secon d - order accuracy (  = 2 ) in FVM , the (dimensionless) firs t moment s must vanish ; (27)    =       ,  (      )     = 0 ,    =       ,  (      )     = 0 ,    =       ,  (      )     = 0 . 5. A small d ispersion e rror: For this goal, i t is necessary to minimiz e t he imaginary part of the transf er function , 13 (28)     (  )   0;  . 6. Positivity : The real part of the tra nsfer functio n must be s trictly positi ve to ensure physical consistency and numerical stability , (29)     (  )  > 0;  . 7. No amplific ation ( filter stabi lity) : The transfer function mag nitude must be less than or equal unity to avoid artificial energy addition and buildup by filterin g , (30)   (  )   1;  . 3.2. The necessity for a new FVM fil ter A class of filte rs i n the context of the FVM for unstructured grids includes differen tial filters . The application of differential fi lters in LES originated with the seminal wor k by Germano ( Germano 1986b, c) who introduced a linear elliptic differential filter. Building on this foundation, diffe rent implicit an d explicit forms of di fferential filters have b een proposed , the former demands a high comput ational cost ( Bose 2012 ) . Here , a widely - used explicit differential fil ter variant , known as Laplace filter, also denoted here as ”laplaceF ilter” , is considered ( Weller et al. 1998 ) : (31)   =  +  . (  ) , w here  is a n adjustable coefficient that determine s the filte r width . An FVM discretized form of the Laplace filter is expressed as : (32)    =   + 1     (  )  (  )  .      ,    , where the sum is over all faces (  ) of cell  ,  is the cell volume,  =    /  is a field assigned to cell centers, ( . )  is the interpolation of a quantity at f ace  ,   is the area v ector of face  , and  is a user -defined constant deter mining the fil ter width . A value of  = 24 corresponds to a grid filte r (  =  ), while a value of  = 6 approximates a filter width o f  =  . F or wall cell faces , the determinatio n of the v eloc ity field gradient — w hich is required in Eq. (32) — is not straightforw ard . The velocity boundary condition at walls is a zero - value d Dirichlet condition . Consequently, the gradient cannot be directly specified and should be either extrapolate d or estimated through ad hoc relations. This introduces inherent uncertai nties, which can compromise the acc uracy and robustness o f simulation s using the L aplace Filter in the vicinity 14 of solid boundaries. T his issue becomes part icularly pronounced in AD - LES with algorithms lik e the Van Cittert method, whe re the filter is app lied iterat ively. More importantly, our channel - flow a posteriori analyses (in section 4 ) demonstrate that the Laplace filter exhibits strong instabilit ies on non - uniform g rids, resulti ng in diverg ence of the solution. To e xplain th e reason fo r this issue, the 3D instantaneous ax ial velocity field of a channel flow at    = 395 , obtained from an LES solution on a typical channel - flow non - un iform Cartes ian grid (with a maximum aspect ratio of 50 and wall - normal grow th rate of 1.1), is considered here as the input (  ) to the 3D explicit filtering operat ion . Figure 3 a presents th e velocity profil e at a channel cross - section along with the corresponding filtered (   ) and deconvoluted (   ) fields profiles, using a Laplace fi lter of width  =  and the Van - Cittert deconvolution algorithm. The deviation (     ) /  is plotted in figure 3 b for various Van - Citt ert iter ations (  ). T he filter is clearly unstable near the wall, as evidenced by the pronounced oscillations in both the filtered and deconvoluted profiles ( figure 3 a) . As shown in figure 3 b , increasing the number of Van - Cittert iterations in tensifies the se oscillatio ns and propagates th em further towa rd s the channel center. Note that a zero - valued v elocity grad ient at the walls, which is common ly used for this filt er, was adopted fo r the results in figure 3 , however, an extrapolated velocity gradient at the boundaries was also tested , which resulted in the same issue. To fu rther support that the se instabilit ies are mainly originated from the grid configur ation rather than the u ncertainty of the Laplace filter at the boundaries , the same velocity profile (  ) is mapped onto a uniform Ca rtesian g rid with a unity aspect ratio , and the computed filtered and deconvoluted profiles on this grid are shown in figure 3 c. The results demonstrate a stable fil ter performanc e, with increasin g iterations re ducing the deviation between   and  ( figure 3 d). Th is critical issue of the L aplace filter under non -uniform grid conditions will be further analyzed based o n the propert ies of filters in the current sec tion . Another lim itation of the stand ard Laplace fil ter formulati on lies in the d efinition of the diffusion co efficient  . As shown in Eq. (32) , while  changes locally with the grid size, it is treated isot r opically as a scalar. This iso tropic defin ition implies that the filte r applies the same diffusion coefficient a cross all face s, rega rdless of the cell asp ect ratio. On highly aniso tropic grids , e.g., boundary layers, this inability to adapt to the grid anisotropy results in excessive filtering or stability issues in the wal l -normal direction. 15 (a) (b) (c) (d) Figure 3: The performance o f the Laplace filt er differential filter on a typ ical channel - flow non - uniform grid (a, b) an d on a uniform grid (c, d): The original (  ), filtered (   ), and deconv oluted (   ) axial velocity profiles at a channel cross - sect ion for  = 2 (a, c) , and the normalize d deviation of the deconvoluted velocity from the original velocity for different  s (b, d). Another discrete filtering approach applicable to general un structured grids is the use of face - averaging method s . A key advantage of thi s tec hnique is th at it does no t require the additional treatment of the velocity gra dient at the boun daries. Furth ermore, it is co mputationa lly efficient , being approx imately five tim es faster than the Laplace filter when applied to the velocity fiel d in a channel flow (based on our computational cost analysis). Among face - averaging methods , the area - weighted face average, known as “ simpleFilt er ” , is the most w idely - used , which is defined by: (33)    =   (  )       ,         ,       (   ) . 16 To analyze the properties of th is filter for a canonical test case, a 3D uniform Cartes ian grid with a geom etric aspect ratio of  , where   =   =    , is considered. It can be shown that t he incorporation of the Laplace and simple f ilter s , i.e., Eq. (32) or (33) , on this grid ( and the use of the standard central 2 nd - order discre tization of (  )  for th e former filter ) leads to the expr essions in the form of Eq. (21) for these filters w ith the coefficien ts,   ,  , given in table 2 . For the sake of comparison , the coeffic ients for a box filter o f width 2 are a lso include d . Additionally, the coefficients for the second - ord er Padé filter ( Lele 1992 ) at a dimensionless cut- off wavenumber of  /2 are reported ; the derivation is pr ovided in Appendix A . A detailed comparison of the f ilter coefficients reveals seve ral key points: 1. Aspect ratio dependence : The coefficients of both the Lapl ace and simple f ilter s exhibit explicit dependence on the grid aspect ratio , i.e., par ameter  . In contrast, the coefficients of the box and Padé filter s are independent of the aspect ratio. 2. Directional variation: For the Laplace and simple f ilte r s , the coefficients in the y -direction are signific antly diffe rent from those in the other directions — specifically, by a factor of   in the case o f the Laplace filter and  in the case o f the simple filter . Th is indicates that the filtering operation is highly anisotropic and predominantly applied in the y -direction. Conversely, the box and Padé fil ters maintain identical c oefficients across all s patial directions. 3. Filter width control: The e ffective width of the L aplace and simple f ilter s varies with th e grid aspect ratio. 17 Table 2: Filter coefficient s on a 3D uniform Car tesian grid with an aspect ratio of  (   =   =    ) for different filter types. Filter coefficients Box filter 2 nd - order Padé filter Laplace filter simple filter   Filter   ,  ,  1 8 1 8 1       4   + 2  1 2 0.5   ±  ,  ,  1 16 1 16 1    1 4 ( 2 +  ) 1 1 2   ,  ±  ,  1 16 1 16      4 ( 2 +  ) 1 1 2   ,  ,  ±  1 16 1 16 1    1 4 ( 2 +  ) 1 1 2   ±  ,  ±  ,  1 32 1 32 0 0 0   ±  ,  ,  ±  1 32 1 32 0 0 0   ,  ±  ,  ±  1 32 1 32 0 0 0   ±  ,  ±  ,  ±  1 64 1 64 0 0 0 To pr ovide a deeper insight into the characteristics of the L aplace and simple f ilter s , we plot their transfe r functions fo r four distinct computational grid configurations. Our analysis begins with unifor m Cartesian grid s . The transfer functions for the Lap lace filter (  = 6 ) and simple filte r under th is condition can be computed by Eq. (22) using the coeffi cients,   ,  , given in table 2 . The magnitude, r eal part, and imaginary p art of the ir transfer function s for a grid with an aspe ct ratio of unity are presented in figure 4 a and b , respectively. Notably, neither filter exhibits a high attenuation at large wavenumbers. To examine the impact of grid anisotropy, fi gur e 4 c and d display the transfer func tions for a u niform grid w ith an aspect ra tio of  = 50 . The simple filter shows a sufficient hig h - wavenumber attenuation only in the wall - n ormal (  ) direction , but this comes at the expense of a signif icant reduction in attenuation in the other two directions . I n t he case of the Laplace filte r ( figure 4 c) , the presence of large negative val ues in the re al part of its transfer func tion in the y - direction on the grid with a high aspect ratio indicates a violation of the positivity property (Eq. (29) ) . Likewise, the transfer function magnitude signifi cantly exceed s unity, demonstrat ing the absence of the filt er stability property (Eq. (30) ). This well explains the instability o f the filter ing operatio n on the h ighly - anisotropic channel - flow grid seen in figure 3 and the divergence of the solution using this filter in our a posteri ori tests (section 4 ) . By the use of the filter stabi lity criterio n , it can be demonstrated that, for a uniform grid with a n aspect ratio of  , t he Laplace fi lter re mains stable on ly when   (  /2 )  /  (the derivation is provided in Appendix B ). For instance, with  = 6 (  =  ) , the maximum permissible aspe ct ratio is 2.2795. 18 (a) (b) (c) (d) Figure 4: The effect of the grid aspect ratio on the magnitud e, real part , and imaginary part of the filte r transfer functio n s on uniform Cartes ian grids : a) The Laplace filter  = 1 , b) simple filter  = 1 , c) Laplace filter  = 50 , and d) simple filter  = 50 . Another repr esentation of the transfe r functions in the range   (1, 50 ) is provided in fig ure 1 of S upplemental Ma terial S1 . In the next step, we proceed to investigate th e propert ies of the L aplace and simple f ilter s under non- uniform Cartes ian and unstru ctured grid conditions. S in ce an analytical determ ination of fil ter coefficients for these scenarios presents a high complexity , we ove rcome this challenge using MATLAB’s ( www.mathworks.com ) Symbolic Math Toolbox. The code to compute the filte r coefficients on these gri ds along with a descrip tion of its alg orithm is giv en in Appendix C . Using the computed filter coefficients on a non - uniform Cartesian grid with a growth rate of 1.1 in the y - direction and a un ity aspect ratio, the transfer func tion ( Eq. (22) ) of Laplace and simple filters is illustrated i n figure 5 a and b , respectively . I t can be observed t hat the real part and the magnitude of the transfer function s have not changed significantly compared to figure 4 a and b . However, the imag inary part of the t ransfer funct ion for both filt ers in the y - direction has become non- zero; for the Lap lace filte r , the maximum magnitude of the imaginary part is 0.05, and for the simple filter , it is 0.025. Therefore, the grid expansion produces a level of disp ersion error (see Eq . (28) ). The mutual effect of th e growth rate and aspect ratio is examined in figure 5 c and d , which 19 respectively display the trans fer function of the Laplace and simple filters for a non - uniform grid with a growth rate of 1.1 and an aspect ratio of 50 ( a typical channel - flow gri d ) . Again, the real part and the magnitude of the transfer function s have not changed significantly compared to the uniform grid in figure 4 c a nd d, however, the ima ginary part has beco me non - zero. With increasing the aspect ratio from 1 to 50, i t is observed that the max imum magnitu de of the ima ginary part o f the transfer fun ction in the y - dire ction increas es for both the Laplace and simple filters, reaching values of 9 and 0.08, respectively. This manifests that the mutual presence of th e aspect ratio intensifies t he impact of t he growth rate. T he fir st moment s of the fil ters are also rep orted in table 3 . Not e that the simple filter is not commutativ e in the dire ctio n of the grid expansion (     0 ) and its deviation from the commutativity property, Eq. (27) , grows with the increase in the aspec t ratio (  ). (a) (b) (c) (d) Figure 5 : The effec ts of the grid grow th rate (  ) an d aspect ratio (  ) on the magnitu de, real part , and imaginary pa rt of the f ilter transfer function s on non - uniform Cartesian grids : a) The Laplace filter (  = 1.1 and  = 1 ), b) simple filter (  = 1.1 and  = 1 ), c) Laplace filter (  = 1.1 and  = 50 ), and d) simple filter (  = 1.1 and  = 50 ). Another representat ion of the transf er functions in th e range   (1, 50 ) is provided i n figure s 2 and 3 of S upplemental Material S1 . 20 Table 3 : The f irst moments of Th e L aplce f ilter, simple filter , and new filter for two non - un iform Cartesian grid s exhibiting a growth rate of 1.1 in the y - direction , with aspect rati os  = 1 and  = 50 .   = 󰇡 (   1 ) 2 +    1  2 + (   1 ) 2 󰇢 1/2 . Grid Properties Laplace filter simple filter   Filter  = 1 and  = 1.1    and    0 0 0    0  0. 0159  0. 0159   0 0. 0159 0. 0159  = 50 and  = 1.1    and    0 0 0    0  0. 0458  0. 0159   0 0. 0458 0. 0159 Based on the observations made and analyses provided in this section , it is concluded that, among properties 2-7 outlined in section 3.1 , The Laplace filter only retains the commutativit y property. While its transfer functio n shape varies across diffe rent direct ions and exh ibits a strong dependence on the computational grid configuration. In contrast, the si mple filter demonstra tes the properties o f positivity , sta bility , and a minimal d ispersion er ror. Neverthe less, simila r to the Laplace filter , its transfer function shape considerabl y changes by direction and is s ignificantly influenced b y the grid st ructure. The se findings collectively suggest tha t neither the L aplace nor simple filter is an ideal candidate for use as an explic it filter in ADM . Both lack th e majority of desirable properties enumera ted in section 3.1 , are highly s ensitive to the underlying computational grid and especially the grid aspect ratio , and do not provide a similar behavior across spatial directions. In light of the se limitatio ns, there is a clear n eed to develop a filter that is ap plicable to general unstructured grids while is , first and foremost, l ess sensitive to grid configuration and ensures isotropic pro perties. More precisely , such a filter sh ould, to the greate st extent poss ible, satisfy the full range of propert ies identified in sect ion 3.1 . 3.3. The new filter definition and analysis To address the c hallenge introduced in section 3.2 , a novel filter is designed that is independent of the grid aspect ratio, eliminates the n eed for ad hoc boundar y treatments , and em ploys a recursive m ethod to dynamic ally adjust the filter w idth. The propo sed filter comb ines a new face - averaging technique with a recursive filtering ap proach. The formu lation of the ne w filter is as follows: (34)     = ( 1    )     +          ;  = 1,2, … ,   ,     =   , 21 (35)   (   ) =  (  )    ,      ,  , where   is t he number of filter recu rsions and   are th e filter relaxation coefficient s, constituting   + 1 filter pa rameters . To justify wh y a new face -averaging,   , is chosen as the base filter instead of   , Eq. (33) , t he characterist ics of   ar e analyzed and compared with the Laplace and simple filters in this section. I n the canonical case of a 3D uniform Cartesian grid with an aspect ratio  , where   =   =    , an alternative form o f the filter r elation for   in Eq. (35) can be expres sed by Eq. (21) with the coefficien ts analytically deriv ed and reported in table 2 . Notably, unlike the Laplace and simple fil ters , the coefficients of the proposed filter are independent of the grid aspect ratio (  ) and remain identical across all spat ial directions. To further a nalyze the base filter, i n figure 6 , the transfer fu nction of th e base filt er (   ) i s examined on four distinct grid confi gurations: a u niform grid with two aspect ratio s of unity and 50, and a non - uni form grid with a growth rate of 1.1 and two aspect ratio s of unity and 50 at the target cell. Overall, the real part and magnitude of the transfer function f or the new filt er exhibit remarkable isotropy across differe nt direction s and remain largely una ffected by v ariations in grid structure. It is also observed that, for both non - uniform grids, the imaginary component of the transfer funct ion in the y - direction r emains identical on both non - uniform grids de spite diff erences in the aspec t ratio, with a maximum value of approximately 0.025 — substantially lower th an th ose of the Laplace and simple fi lters ( 9 and 0.08 ). Furthermore , the first moment of   (reported in table 3 ) re mains much sma ller than tha t of the simple filter on a typical channel - flow gr id configuration (  = 1.1 and  = 50 ). Nevertheless, the high - wavenumber attenuation property is not fully achieved by this filter , simila r to the Laplace and simple filters . To achieve improved filters in te rms of satis fying the se t of filter p roperties de fined in sect ion 3.1 and more import antly to be able to adjust a prescribed filter width, an optimization prob lem is formul ated next to determine th e filter parame ters,   and   (  = 1,2, … ,   ) . 22 (a) (b) Figure 6 : The effec ts of the grid grow th rate (  ) an d aspect ratio (  ) on the magnitude, re al part, and imaginary part o f the base filter (   ) transfer function on different Cartesian grids: a)  = 1 and  = 1 or 50 , b)  = 1.1 and  = 1 or 50 . 3.4. The new filter o p timization procedure In this section, the parameters of the new filter , including the number of recursions,   , and relaxation coefficients,   (  = 1,2, … ,   ) , are determined based o n a multi -o bjective optimiza tion process to achieve the des ired explicit fil ter properti es introduced in section 3.1 . T he normaliz ation property , Eq. (24) , is aut omatically s atisfied by t he definition of the new filter , E qs. (34) and (35) , and thus is not included explicitly as an optimization goal . The n ormalized obje ctive functions and constraints of the optimization problem are defined in table 4 . Note that, here, the wavenumber space is discretized into ten equidistant points in each sp atial direction within the range ( 0,  ) , indexed by  = 1, … , 10 . These point s are used to compu te the representative va lues of the transfe r function in the wavenumber space. 23 Table 4 : Definition of the normaliz ed objective f unctions and cons traints for the optimiza tion problem.   and   are the unit vector and filter width in direction  . Type Description Mathemati cal Formul ation Count Objective : min {   } High - wavenumbe r attenuation: Based on E q. (25) minimizes th e transfer function norm at the N yquist wavenum ber (  ).   =   (  =    )       /  ( 36 ) 1 Objective : min {   } Commutativity: Based on E q. (27) minimizes th e norm of the first - order mom ents to ensure com mutati on with spatial derivat ive opera tors.   = 󰇡 (    )  +      + (    )  󰇢  /  ( 37 ) 1 Objective : min {   } Small dis persion error: Based on E q. (28) minimizes th e imagina ry part of the tra nsfer function across 10 representative points per directi on.   = 󰇭       =  10         󰇮  /  ( 38 ) 1 Constrai nt C1: Preadjusted filter width: Constrai ns the m agnitude to ~ 0.5 at the specific cut - off frequency   =    /   . 0 . 495     =          0. 505  = 1,2,3 ( 39 ) 3 Constrai nt C2: Positiveness: Ensures the rea l part of the tra nsfer function remains positive at represe ntative poi nts.      =  10    < 0  = 1,2,3;  = 1,2, … , 10 ( 40 ) 30 Constrai nt C3: Stability: Ensures the tra nsfer fu nction magnitude does n ot excee d unity (no amplification) at representative points. 󰈅   󰇧 =        󰇨󰈅  1  = 1,2,3;  = 1,2, … , 10 ( 41 ) 30 The optimization problem is thus defined by th re e objective functi on s   ,   , and   , along with 63 nonlinear constrain ts   ,   , and   . The decision variables include   and   (  = 1,2, … ,   ) , resulting in a total of   + 1 decision parameter s. Here,   is chosen in the range of 3 and 4 (based on comp utational effic iency considera tions ) , and the range of relaxation   is constrained to be [0,1. 5] ( based on prior testing and p ractical experien ce, as it ensures filte r stability ). Given that   is restricted to discrete values , the optimiza tion problem is solved independently for each   value, considering   s as the d ecision variables only, and the most optimal solu tion is select ed from the op timum s for all   . 24 To solve this multi - objective opti mization prob lem, MATLAB ( www.mathworks.com ) a nd the “gamultiobj” function, using a m ulti -o bjective Genetic A lgorithm (GA) ( Deb 2001 ) , we re employed. This approach is particularly well - suited for problems involving multiple competing objective fu nctions, as it eff iciently e xplores the Pa reto front to id entify optima l trade -offs. W e share the optimization code and an example on how to use it for a non - uniform Carte sian grid typical in channel flow problems in Supplementary Material S3 . The readers can simply ado pt it for their ow n grid. To systemat ically select th e optimal solution in a multi - objective optimiz ation framework, we employ a minimu m distance - based decision - making function. Given three objective functions   (  = 1,2,3) and an ideal solution at    = (0,0,0) , the decision fu nction is formul ated to minimize the weighted Euclide an distance from this ide al point. Specifically, w e define the deci sion -making function as follows: (42)  =   (       )        /  , where   represents the assigned weight for each ob jective, ensuring a balanced trade - of f among conflicting criteria. In this study, we assume equal importance of objectives, i.e.,   = (1,1,1) , while other weights can be simply adopted by the users. 3.5. The new filter o ptimization results The ALDME mode l necessitates a filter width of two times the grid size , serving as both the explicit convolution (. )     and test (. )  filters. Therefore, he re we design an expl icit filter of width   /   = 2 , necessary for the imple mentat ion of the AD - LES model describe d in section 2 . Nevertheless, the design algorithm and provided code described in section 3 can also be employ ed for other fi lter widths. I t is important to note that the transfer function depends not only on the filter parameters but also on the ch aracteristics of the computationa l grid. I n wall - bounded flows, the grid is typically re fined near the wall and co arsens at a specified growth rate as the distance from the wall inc reases. T he optim ization is performed on a non- uniform Cartesian grid with a growth rate of 1.1 in the y - dire ction and an aspect ratio of 50 near the walls , a typical choice in LES applications ( Gullbrand 2004 , Sagaut 2005 , Choi & Moin 2012 , Singh & You 2013 , Schumann et al. 2020, van Dr uenen & Blocken 2024 ) . The new filter coefficients are independent of the grid aspect ra tio (see section 3.3 ), th erefore, thi s grid config uration ens ures that th e 25 optimized f ilter perfor ms effective ly across th e range of scales e ncountered in practica l simulations. The results of the optimization proc ess are summarized in tab le 5 . Additionally, the magnitude, real , and imagina ry parts of th e transfer fun ction s for filter s with   values of 3 and 4 are illustra ted in fi gure 7 . As eviden t from the results , the transfer fu nction s for both   values exhibit nearly identical characteristics . Therefore , when considering the computational cost, the most opti mal filte r is ach ieved by choosing   = 3 and   = [1. 2117 , 1. 2344 , 1. 2189 ] . It should be noted that th e result of a GA optimization is prone to a level of variation and uncertainty, depending on the initializat ion, chosen algorithm parameters ’ values , r andomness of different step s, and the complexity of the problem. To ensure the validity of the optimizati on results, we conduct ed an u ncertainty analysis by repeating the optimiza tion process 100 independent times. The filter coefficients (   ) showed a minor variation with a standard deviation of approximately 0.02 (representin g a variation of  1. 7% relative to the mean) , and the objec tive function values ( O  , O  , O  ) exhibited standard dev iations in th e order of 10  to 10  . Table 5 : The opti mal filter parameters o f the new filt er for a typical chann el - flow grid (a no n -uniform Cartesian grid with  = 1.1 and  = 50 ).                 3 1.2117 1.2344 1.2189 - 0.3607 0.0583 0.0687 4 0.9581 0.9719 0.9225 0.9182 0.3830 0.0600 0.0662 (a) (b) Figure 7 : The magnit ude, real pa rt, and imagi nary part of the t ransfer func tions of the opti mized new f ilter correspondin g to the filter parame ters given in table 5 : a)   = 3 and b)   = 4 . It is worth noting that the divergence - preserv ing property of an explicit filter math ematically relies on its co mmutativity (                  ), which was accounted in this work. However, another 26 condition on the sum of fluxes at the faces of computational c ells ( Agdestein & Sanderse 2025 ) is also necessary to ensure the divergence -preserving property (       = 0 ), which was not considered here. 3.6. The new filter optimization for tetrahedral grids Sin c e the new filte r is applicab le to general un structured g rids, we invest igate the filte r parameters and performance for general tetrahedral unstructured grids in this sect ion. To assess the performance of the new filter on unstructured g rids, its transfer fun ction is evaluat ed over two tetrahedral grid configurations. The first grid (shown in figure 8 a ) exh ibits an aspect ratio o f 1.23, a mean volume ratio of 1.25, and a skewness equi angle of 0.386, while the second grid (shown in figure 8 b ) whi ch resembles a boundary layer grid has an aspect ratio of 5.68, a mean v olume ratio of 1.34, and a skewness equiangle of 0.934. For this analysis, the optimization (for   = 3 ) yields the filter re laxation pa rameters of   = [1. 0786 ,1. 0765 ,1. 0829 ] for the first grid and   = [0. 9968 ,1. 0130 ,1. 0199 ] for the second one. The MATLAB opti mization code, the inputs and settings for these two unstructured t etrahedral grids, and a succinct user guide are available in Supplementar y Material S4 . This resource includes instructions for utilizing the code as well as guidan ce on executing the scripts . Figure 9 a and b present the transfe r functions of the filter s with the optimized coefficients. Notably, despite the relatively low quality o f the comput ational grid, the filter m aintains clos e to isotropi c, except for the dispersio n error, with a reasonabl y transfer fun ction properti es across each spatial direc tion . (a) (b) Figure 8 : Visualization of the two unstr uctured tetrahed ral grids: a) Th e first grid with an asp ect ratio close to unity , and b) the s econd grid wi th a large as pect ratio. 27 (a) (b) Figure 9: The magnitude , real par t, and imagina ry part of the o ptimal new f ilter trans fer function over : a) the first t etrahedral grid, and b) the s econd tetra hedral grid. 4. The a posteriori analysis of the new filter This section evaluate s the performance of the proposed filter for AD - LES using a general unstructured grid code, i.e., OpenFOAM. 4.1. New filter performance on highly stretched boundary- layer grids In the chose n AD - LES closure, i.e., ALDM E (see section 2 ) , both the deconvolution f ilter and the test filter ha ve a width of 2  󰆻 =  , where   =  =    and  is the local cell volume. Consequently, the filter paramet ers obtained from the optimization process in s ection 3.5 — specifically,   = 3 and   = [1. 2182 , 1. 2088 , 1. 2381 ] — are adopted for the current simulations . For each of the chann el flow benchmarks at    = 395 and    = 5 90 , t h ree AD - LES simulations are carried out , each usin g one of the FVM e xplicit filters investigated in this work fo r general unstructured grids, i.e., The Laplace f ilter , simple fil ter , and newly developed filter (newFilter) . A ll simulation s using the Laplace fil ter diverged. Just before divergence, strong unphysical oscillation s of the velocity w ere observed near the wall boundaries, similar to those occurred in our a priori analysis (see figure 3 ). Th e rea son behind this issue of the L aplace filter was analyzed in detail in section 3.2 . Figure 10 and figure 11 present the flow statistic s , inc ludi ng the mean velocity and Reynolds stress profiles, pr edicted using t he new filter a nd widely - used simple filter ag ainst the reference DNS data. The results using the new filter demonstrates noticeable improve ments compared to the one with the simple fil ter , specifically for the mean velocity pro file. The a mount of log - layer mismatch is allev iated by the use of the new filter. It is 28 worth notin g that some previous studies, e.g., Schlatter et al. (2004) , often repor ted limited improvements with conventional AD - LES closures compared to pure DEV - LES. Through theoretical an alyses in our recent work ( Amani et al. 2024) , we demonstrated that existi ng conventional penalty - term AD - LES and even standard mixed AD - LES suffer from in herent mathematica l inconsistencies. Then, we proposed novel mixed AD - LES closures and proved their advantages in several a posteriori test s. The consistent mixed approaches can combine the best of both worlds: the high - fidelity struct ural reconstru ction of AD and the stability of E V , yielding results that are superior to conventional methods. From those consistent mixed AD - LES models, the best one has been chosen in the present wor k to investigate the ef fect of explicit filter design. The result of a widely - used pure EV - LES, namely WALE, is also reported in figure 10 a . Th e comparison provided i n this figure highligh ts that the co nsistent mixe d AD - LES model improve s the flow predictions, more noti ceably when a proper explicit filter such a s the one designed in the present work is adopted. In other word, the explicit filter design is the key element of AD - LES to achieve high fidelity resu lts . (a) (b) Figure 10 : The (dimension less) mean stream wise velocity profiles: A comp arison of th e AD - LES results using the n ew and simple filter ag ainst the DN S data ( Moser et al. 1999 ) : (a)    = 395 , and (b)    = 590 . The AD - LES u sing the Laplace fil ter diverged . Part (a) includ es the results of t he pure eddy - viscosity WALE as a r eference LES . As shown in figure 11 , using the n ew filter, i mpro vements are also observe d in the prediction of the strea mwise Reynolds stress , which is the dominant no rmal Reynolds stress component , and the shear stress , specifically at higher Reynolds numbers (    = 590 ). However, a level of under- prediction in the wall - normal Reynolds stress co mponent ,    󰆒   󰆒  , is evidence d with the ne w filter at the higher Reynolds number ( figure 11 d ). Notably , as shown by Fröhlich and Rodi (2002) , the high over - prediction of the peak    󰆒   󰆒  value ( figure 11 a and b ) in LES results of chann el flow 29 may be connected to the grid size in the spanwise direction, and the redu ction of this factor can effectively mitigate this error. (a) (b) (c) (d) Figure 11 : The (dimension less) Reynol ds stress es : A comparison of AD - LES re sults using the new and simple f ilter s against the DNS data ( M oser et al. 1999 ) : Left column:    = 395 , and right column:   = 590 . The AD - LES using the L aplace f ilter diverged. To explore the origin of these enh ancements using the new filt er in greater det ail s , the components of the SFS stresses , i.e.,   and   in Eq. (6) , are plotted in figure 12 . According to these figures , using the new filter lea ds to a significant inc rease in the values of these two stresses. For a more quanti tative analysis , tabl e 6 reports the contribut ions of various terms to the turbulent kinetic energy. The data in this table further support the earli er observation o f increased SF S stresses . On the other hand, a pplying the new filter results in approximately a 30% reduction in the resolved kinetic ene rgy. This is justified by the comparison of the shap e of the trans fer functions of the new filter ( figure 7 a) and simple filter ( figure 5 d). There is an insufficient level of attenuation near the Nyquist in the x - and z - directions using the simple filter . O n the other hand, 30 the new filter operates isot r opically and offers a reasonable attenuation in all three d irections. As a result , the level o f directly re solved kineti c energy declines and the resolv ed part using deconvolution increases using the new filter . Therefore , i t is inferred that filterin g out the frequencies near the Nyqu ist (sufficient attenuation near the Nyquist) is an import ant feature of a n ADM explicit filt er , which is effectively achieved in all directions by the new filter and brings about more accura te pred ictions . It is worth m entioning tha t almost al l of this redu ction in the directly reso lved kinetic energy is recovered by the deconvolution (deconvolved kinetic energy in table 6 ) and the fraction of the modeled kinetic energy still remains below 10% for the new filter in both cases , reflecting well - resolved LES based on the wel l -known Pope’s criterion ( Pope 2000 ) . To further show that the incor poration of the new filter in our study meets the c riteria for a high - quality LES, the LES quality criterion based on the index of resolution ( Celik et al. 2005) , namely the ratio of the eddy viscosity to the molec ular viscosity, is examined in figure 13 . It can be seen that this r atio is well b elow the cri tical value o f 20, for a we ll - resolved LES, in both benchmark cases using the new filter . Figure 12 : A comparis on of the deconvol ved ,   , (top row) a nd modeled,   , (bottom row) SFS stress components u sing the new and simple filter s. 31 Table 6: A comparison of the kine tic energy bu dget ( in percent) of the directly resolved, decon volved, and modeled stress using the n ew and simple filter s. Benchmark Filter 1 2    󰆒     󰆒    ( Directly r esolved ) 1 2     (Deconv olved)   (Modeled)    = 3 95 simpleFilter 81 . 83 15 .2 2. 97 newFilter 49 . 17 43 .7 7. 12    = 5 90 simpleFilter 85 . 62 12 . 02 2. 36 newFilter 56 . 22 38 . 15 5. 63 Figure 13 : A comparis on of the eddy viscosity ratio profiles using the new and simple filter s. 4.2. New filter perform ance on unstructured grids For a posteriori tests of unstructured grids in the TGV benchmark, as a simplific ation, the opt imal parameters obtained in section 3.6 — specifically,   = 3 and   = [1. 0786 ,1. 0765 ,1. 0829 ] — are used in this section . Th e computational c ell size of the re ference D NS solution (DeBonis 2013) is denoted by   . For the cu rrent LES, un structured g rids ( figur e 2 ) composed of prismatic c ells (with hexagonal bases in the x - y planes and prism axes in the z - direction) and different aspect ratios are generated to study this benchmark. The cell base siz e is indicated by   and the height size by   . Note that the prism c ell topology i s used in pl ace of a tetrahedral one, since the former is more suite d for LES , especially using stretched grid cells, to avoid too larg e dissipation errors. Figure 14 shows iso - surfaces of the Q - criterion predicted by the present AD - LES on an unstructured grid. Figure 15 presents a comparison of the resu lts of AD - LES on unstructured grids with differe nt cell aspec t ratios , using bo th the simple and new filters , against DNS data . For the grid with a low aspect ratio (   /   = 4,   /   = 2 ) and the highest resolution close to the 32 DNS, the pr edicted kine tic energy e volutions in time are in close agree ment with the DNS . This highlights the consistency of the solver and solution, using both filters, in the limit where the resolution approaches to the one of DNS. It should be em phasized that while the LES grid resolution is very close t o the DNS in this case, the la tter possesses higher spa tial and temporal discretization schemes . As the grid resolution decrea ses, by increasing the cell aspect ra tio (stretch), the deviations of the LES results from t he DNS data grow for b oth filters. How ever, at the highest value of th e stretch (   /   = 16 ,   /   = 2 ) , the result using the new filter demonstrates a much closer agreement w ith the DNS in the seco nd half of the simulation period, while in the first ha lf, the results of both filters a re very close togethe r. This can be justified by noting that the overall size of the vortical s tructures , in th is decaying b enchmark , reduces by time (see figure 1 4 ) . Considering the smaller vortical structure s in the second half of the simulation, the effects of the filte r cut - off length and properties are deemed much more important in the second period. Finally , it is w orth noting that in this section, as a simplificat ion, the optim al parameters obtained for tetrahedral c ells are used for pri sm cells. Th is simplifi cation is made owing to the low sensitivity o f the filter coeffici ents observed in section 3.6 . Further examination of this simplification and obt aining the filter co efficients for differen t cell topolog ies would be a valuable topic for fut ure research. (a) (b) Figure 14 : The TGV benchmark: Q- crit erion iso - surfaces predict ed by the p resent AD - LES on an unstructure d grid (   /   = 4,   /   = 2 ) at time   /  = 5 (a) and 15 (b). 33 Figure 15 : The TGV benchmark: T he (dimensionl ess) volume - av eraged total kinetic ener gy versus dimensionle ss time. T he comparison of the DNS soluti on ( DeBonis 2013 ) with the present AD - LES results us ing diffe rent filters — Simple Filter (SF) and New Filter (NF) — and grid aspect rat ios: low stretch (   /   = 4,   /   = 2 ), medium stretch (   /   = 8,   /   = 2 ), high stretch (   /   = 16 ,   /   = 2 ). A zoomed - in view around   /  = 15 is shown in t he inset. 5. Conclusion This study a imed to develo p and evalua te a novel opt imized recurs ive explic it filter for AD - LES applicable to unstructured grids . To this end, detailed analyses were first conducted on two commonly used fil ters for genera l unstructured grids — namely, a differe ntial filter (l aplaceFilter ) and a face - averaging fil ter (simpleFil ter) — across various canonical test cases. It was demonstrated that both filters exhib it strong dire ctional an isotropy in the ir transfe r function s hape and ac ute performance sensitivity to grid configuration, especially to the cell aspect ratio. Both filters showed inadequate attenuation at high wavenumbers in all or some spatial directions on typical channel - flow grids. In addition, the Laplace filter violate s the filter sta bility and po sitiv ity cri teria on grids with moderate to high aspect ratios, eventually leading t o solution divergence. On the other hand, for the simple filte r , the co mmutativity erro r grow s with the increas e in the aspect ratio. To address these shortcomings , we proposed a new f ilter , first and foremost, to remove the strong sensitivity of the filte r coefficient s and transf er function to the grid aspect ratio and deliver a n isotropic transfer func tion. For this purpose, a recursive filtering approach and a face -averaging technique w ere combined. Then, through a constrained multi - objective opt imization, the filter p arameters were determined to precisely ad just a prespeci fied filter widt h and maintain the fi lter stability a nd positivity , w hile maximi zing the high - wavenum ber attenuation and minimizing the dispersion and commutativi ty errors . Comparative a priori analyses in several tes t cases, including uniform 34 Cartesian grids of different a spect ratios, no n - uniform Cartesia n grids of various aspect ratios, and unstructured tetrahedral grids with small a nd large aspect r atio s , reveal ed substantial improvements in the properties o f the new filter ov er those of th e conventional filters. Furthermore, the proposed filter obviate d the need for additional boundary treatments and offer ed much more parallel co mputational efficiency compared to diffe rential filter s . Finally, we conducted a posteriori analyses through AD - LE S of two turbulent channel flow benchmarks . While all simulations satisfied th e criteria o f a well - resolved LES, t he mean velocity prediction using the new filter sh owed noticeable imp rovements over those using the other filters , and there w ere also slight enhancement s in the prediction of the normal st reamwise and shear Reynolds st ress components. This progress was primarily linked to the suf ficient and iso tropic high-wavenumber attenuating property of the new filte r for an AD - LES . Finally, to show the application of the new filter on more general unstructured grids, a posteriori tests on the 3D TGV benchmark using prism cells were carried out successfully. Future directions for extending the present study include further enhancement s to the filter properties, e.g., by accounting for the divergence - pre servi ng property, and the test of the present filte r performance on different unstructu red - grid cell topologies, which necessitates the use of the proposed algorithm to obtain the optimi zed filter coefficient s and the design of unstru ctured - grid a priori or a posteriori benchmarks. Supplementa ry materials The supplemen tal materia ls include additional graphs of fil ter properti es ( Supplementary material S1 ), the computer programs and user guides for t he calculation of the filter coefficients on arbitrar y grids ( Supplementary material S2 ), t he optimization co de for a non - uniform Cartes ian grid ( Supplement ary material S3 ) , and t he optim ization code for unstructu red tetrah edral grids ( Supplement ary material S4 ). Data availa bility The data that support the findings of this study are available from the corresponding author upon reasonable request. Declaration of Interests: The autho rs report no conflict of interest 35 Acknowledgments This work was partially supported by T Ü BITAK [grant number 221M421]. Appendix A : The 2 nd - order Padé filter coefficien ts The general one - dimensional form of the second - order Padé filter ( Lele 1992 ) on a uniform grid is expressed as follows: (43)     +    +     =    +  2 (   +   ) , where  =    cos (   ) and  =  =   +  . For a non - dimensional cut - off wavenumber of   =  /2 , Eq. (43) i s simplifie d to: (44)    = 1 2   + 1 4 (   +   ) . When this one - dimensional filter is applied independently along each of the three principal directions on a uniform Cartesia n grid , the resul ting coefficients are id entical to those present ed in table 2 . Appendix B : Stability criter ion of the Laplace filter on uniform Cartesian grids of different aspect ratio s To assess the stabili ty of the Laplace fil ter on a uniform Ca rtesian grid, its transfer function is analytically derived in the  ,  , and  directions, and its stability is exa mined based on Eq. (30) . Specifically, the transfer function given by Eq. (22) is expanded for each spatial direction . For instance , the transfer fun ction in the x -direction becomes : (45)    =    =      , 0,0  =    ,    󰇡     ,  ,  󰇢 .(     ) =    =   ,  ,  +   ,  ,          +   ,  ,          +   ,   ,  +   ,   ,  +   ,  ,   +   ,  ,   . S ubstituting the filter c oefficients ,   ,  ,  , on a uniform grid with an aspect ratio of  in the y - direction from table 2 , E q. (45) is reduced to (46)    = 1 + 2     ( cos    1 ) . Applying th e stability crite rion , Eq. (30) , yields: (47)  1  󰇩 1 + 2     ( cos    1 ) 󰇪  1. 36 Since  and  are always positive, the second inequality is a lways held for all   while the firs t one is true if (48)      2. If  > 2 , the above condition is always met si nce  > 1 is assumed . By a simil ar argument, it can be shown that for    , the stability condition is the sa me as Eq. (48) . Applying a simi lar approach for th e transfer function in the y- direction ,    , yie lds t he following constraint: (49)      2    󰇡  2 󰇢   . Based on Eq. (49) , the Laplace filter with a coefficient of  is stable only when the aspe ct ratio of the grid is less than or equal (  /2 )   . Appendix C : The c alculation of the filter coe fficients on arbitrary grids For the calculation of the filter coe fficients on a rbitrary grids— specifically te trahedral and hexahedral grids — computational grids dat a were first imported in to MATLAB. A symbolic field was then constructed to represent a field values at each computational cell, upon which the filtering operation was applied symbolically to the target cell. 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