Deconvolution-based nonlinear filtering for incompressible flows at moderately large Reynolds numbers

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1 Numerical Analysis and Scientific Computing Preprint Seria Deconvolution-based nonlinear filtering for incompressible flows at moderately large Reynolds numbers L. Bertagna A. Quaini A. Veneziani Preprint #37 Department of Matematics University of Houston April 2015

2 Deconvolution-based nonlinear filtering for incompressible flows at moderately large Reynolds numbers L. Bertagna 1, A. Quaini 2, A. Veneziani 1 1 Department of Matematics and Computer Science, Emory University, Atlanta (GA) USA 2 Department of Matematics, University of Houston, Houston (TX) USA SUMMARY We consider a Leray model wit a deconvolution-based indicator function for te simulation of incompressible fluid flow at moderately large Reynolds number (in te range of few tousand) wit underresolved meses. For te implementation of te model, we adopt a tree-step algoritm called evolve-filterrelax (EFR) tat requires (i) te solution of a Navier-Stokes problem, (ii) te solution of a Stokes-like problem to filter te Navier-Stokes velocity field, and (iii) a final relaxation step. We take advantage of a reformulation of te EFR algoritm as an operator splitting metod to analyze te impact of te filter on te final solution vs a direct simulation of te Navier-Stokes equations. In addition, we provide some direction for tuning te parameters involved in te model based on pysical and numerical arguments. Our approac is validated against experimental data for fluid flow in an idealized medical device (consisting of a conical convergent, a narrow troat, and a sudden expansion, as recommended by te U.S. Food and Drug Administration). Numerical results are in good quantitative agreement wit te measured axial components of te velocity and pressures for two different flow rates corresponding to turbulent regimes, even for meses wit a mes size more tan 40 times larger tan te smallest turbulent scale. After several numerical experiments, we perform a preliminary sensitivity analysis of te computed solution to te parameters involved in te model. Received... KEY WORDS: Computational incompressible fluid dynamics; Leray model; nonlinear filtering; approximate deconvolution; experimental validation 1. INTRODUCTION Te Incompressible Navier-Stokes equations (INSE) feature several callenging aspects, e.g. te saddle-point nature of te problem and nonlinearity, tat make teir analysis difficult [48]. Wen numerical metods are used, tese callenging features may result in restriction to te possible discretization settings. For instance, te saddle-point nature of te problem leads to te well-known discrete inf-sup (or LBB) condition [12], wic enforces a restriction on ow to approximate fluid velocity and pressure. Additional difficulties are peculiar to te discrete framework, suc as instabilities in convection dominated flow. Tese can be avoided by restrictions on te discretization parameters or by adopting suitable stabilization tecniques (e.g., [13]). As well known, from te penomenology point of view, strong convective fields compared wit viscous forces may trigger flow disturbances up to turbulence. Tis introduces oter numerical callenges. Wen te Reynolds number - te dimensionless number tat weigs te importance of inertial forces vs viscous ones - increases, te flow is caracterized by te presence of flow structures Correspondence to: 4800 Caloun Rd. Houston, TX, 77004, quaini@mat.u.edu Current affiliation: Department of Scientific Computing, Florida State University, Tallaassee (FL) USA

3 2 L. BERTAGNA ET AL. over a large variety of space scales. Numerical discretization needs to capture all tese structures to give an accurate description of te flow. Unfortunately, as te Reynolds number increases, tis demands for very fine space reticulations tat quickly lead to large algebraic systems. Terefore, tis approac (Direct Numerical Simulation - DNS) may be computationally unaffordable. A possible way to keep an affordable size of te discrete problem witout sacrificing te accuracy is to solve te flow average and model properly te effects of te small scales (not directly solved) at te medium and large scales. Te Navier-Stokes equations can be averaged in time, leading to te so called Reynolds-Averaged Navier-Stokes equations (RANS), or in space, leading to Large Eddy Simulation (LES) tecniques (see, e.g., [17]). In tis article, we focus on te latter approac. We consider a variant of te so called Leray model [33], were small scale effects are described by a set of equations to be added to te discrete INSE formulated on te unresolved mes. Te extra-problem can be devised in different ways, for instance by a functional splitting of te solved and unresolved scales [5] or by resorting to te concept of suitability of weak solutions [26]. Te variant of te Leray model we deal wit was originally proposed in [9]. Here te extra problem acts as a differential low-pass filter. For its actual implementation, we use te evolve-filterrelax (EFR) algoritm presented in [32]. One of te advantages of tis approac is tat it is easily implemented in a legacy Navier-Stokes solver, since te filtering step requires te solution of a Stokes-like problem. We target specifically applications involving incompressible fluid flow at moderately large Reynolds numbers (up to 5000). We reformulate te metod of [9, 32] in an operator splitting framework for te INSE. In addition, we provide practical directions to tune te parameters involved in te model based on pysical and numerical arguments. We test our reformulation and te parameter tuning on a realistic 3D problem, i.e. flow at different Reynolds numbers troug a nozzle wic contains all te features commonly encountered in medical devices (flow contraction and expansion, recirculation zones etc., see Figure 1). We selected tis problem because it is part of a bencmark issued by te U.S. Food and Drug Administration (FDA) witin te Critical Pat Initiative program [2]. Tree independent laboratories were requested by FDA to perform flow visualization experiments on fabricated nozzle models for different flow rates up to Reynolds number 6500 [27]. Tis resulted in bencmark data available online to te scientific community for te validation of Computational Fluid Dynamics (CFD) simulations [1]. Available experimental measurements enable us to ceck te effectiveness of te EFR algoritm in simulating average macroscopic quantities. Before te present paper, te EFR algoritm as been applied to academic problems. Here, it is applied for te first time to a realistic problem of practical interest, featuring non-diriclet boundary conditions, as usually is te case in engineering applications. Results of a first CFD study of te FDA nozzle model are reported in [45]. Several groups of CFD professionals participated in te study, following different modeling approaces (turbulence models vs. direct numerical simulations, coice of te boundary conditions, etc.). Overall, te results obtained by different groups ad a very large variability also wit respect to te experimental results. It was observed tat RANS turbulence models were in general unable to correctly estimate te centerline velocities in te troat of te nozzle and downstream of te sudden expansion. None of te participants in te study used a LES approac. Tanks to EFR algoritm and an appropriate coice for te parameter in te Leray model, we manage to acieve good agreement between te computed velocities and pressures and te respective measured quantities, even for meses wit a size more tan 40 times larger tan te smallest turbulent scale. Moreover, troug a large set of numerical experiments, we sow te impact of te parameters involved in te model on te computed solution. Te paper is organized as follow. In Section 2, we introduce te continuous Leray model as well as te numerical approximation proposed in [32]. In Section 3, we present possible coices for te so called indicator function, a fundamental component of te Leray model. In Section 4, we formulate te algoritm as an operator splitting metod. We consider specifically practical problems wit any kind of pysically relevant boundary conditions. In addition we sow ow certain parameters of te model can be related to pysical and discretization quantities. In Section 5, we describe te details of te discretization and te preconditioners used to solve te resulting linear systems. 2

4 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 3 Te numerical results for te FDA bencmark and te comparison wit te experimental data are reported in Section 6. Conclusions are drawn in Section 7. Software and implementation All te computational results presented in tis article ave been performed wit LifeV [3], an open source library of algoritms and data structures for te numerical solution of partial differential equations wit ig performance computing tecniques. In [37], we validated one Navier-Stoker solver implemented in LifeV wit DNS up to Reynolds number 3500 and sowed tat a properly refined mes is able to capture accurately te average flow features observed in te experiments. Te LES tecnique used in tis article makes te simulation of iger Reynolds numbers computationally affordable witout sacrificing accuracy. An important outcome of tis work is tat te code created for it is incorporated in an open-source library and terefore is readily sared wit te community Te Navier-Stokes equations 2. PROBLEM DEFINITION We consider te motion of an incompressible viscous fluid in a time-independent domain Ω over a time interval of interest (t 0, T ). Te flow is described by te incompressible Navier-Stokes equations endowed wit te boundary and initial conditions ρ t u + ρ (u )u σ = f in Ω (t 0, T ), (1) u = 0 in Ω (t 0, T ), (2) u = u D on Ω D (t 0, T ), (3) σ n = g on Ω N (t 0, T ), (4) u = u 0 in Ω {t 0 }, were Ω D Ω N = Ω and Ω D Ω N =. Here ρ is te fluid density, u is te fluid velocity, t denotes te time derivative, σ is te Caucy stress tensor, f accounts for possible body forces (suc as, e.g., gravity), u D, g and u 0 are given. Equation (1) represents te conservation of te linear momentum, wile eq. (2) represents te conservation of te mass. For Newtonian fluids σ can be written as σ(u, p) = pi + µ( u + u T ), (5) were p is te pressure and µ is te constant (dynamic) viscosity. Let us introduce operators s and s, defined as s u 1 2 ( u + ut ) s u s u. (6) were s is te symmetric part of te gradient. For completeness, we also define operator ss u u s u 1 2 ( u ut ), (7) tat we will use later on. Te incompressible Navier-Stokes equations can be rewritten as ρ t u + ρ (u )u + p 2µ s u = f in Ω (t 0, T ), (8) 3 u = 0 in Ω (t 0, T ). (9)

5 4 L. BERTAGNA ET AL. Remark 2.1 In te continuous formulation, u T = ( u) = 0, due to te continuity equation. Terefore, te Navier-Stokes equations are often formulated wit te operator rater tan s. However, te contribution of te term u T does not vanis wen problem (1)-(2) is formulated in its weak form - as done in te finite element approximation [25]. We retain terefore tis term for te formal correctness of our formulation. Tis term is particularly important e.g. in Fluid-Structure Interaction problems were we ave continuity of te normal stress at te interface between fluid and structure. To caracterize te flow regime under consideration, we define te Reynolds number as Re = UL ν, (10) were ν = µ/ρ is te kinematic viscosity of te fluid, and U and L are caracteristic macroscopic velocity and lengt respectively. For an internal flow in a cylindrical pipe, U is te mean sectional velocity and L is te diameter. For large Reynolds numbers, inertial forces are dominant over viscous forces and vice versa. For moderately large Reynolds numbers te effects of flow disturbances cannot be neglected, and yet Reynolds-averaged Navier-Stokes (RANS) models [38] are generally too crude Leray model To motivate te introduction of te Leray model, it is useful to look at te beavior of te turbulent kinetic energy (TKE) of te fluid, wic is te kinetic energy associated wit eddies in te turbulent flow. In te framework of te Kolmogorov 1941 (K41) teory [31, 30], te TKE is injected in te system at te large scales (low wave numbers). Since te large scale eddies are unstable, tey break down, transferring te energy to smaller eddies. Finally, te TKE is dissipated by te viscous forces at te small scales (ig wave numbers). Tis process is usually referred to as energy cascade. Te scale at wic te viscous forces dissipate energy is referred to as Kolmogorov scale and can be expressed as ( ) ν 3 1/4 η =, (11) ε were ε is te time-average of te rate at wic te energy is dissipated (see e.g. [20]). Formally, ε is defined as 1 T ε := lim sup ν u 2 L T T Ω 2dt. t 0 For a flow in developed turbulent regime, te dissipation rate as to be of te same magnitude of te production rate, wic is te rate at wic te TKE is supplied to te small scales. A common way to express ε in terms of te macro-scale variables is ε U 3 /L [49], leading to te expression η = Re 3/4 L. (12) Tis scaling law pinpoints te difficulty of te numerical solution of te Navier-Stokes equation at ig Reynolds numbers. In order to correctly capture te dissipated energy, DNS needs a grid wit spacing η. As te Reynolds number increases, DNS leads to a uge number of unknowns and proibitive computational costs. On te oter and, wen te mes size fails to resolve te Kolmogorov scale, te under-diffusion in te simulation leads to nonpysical computed velocities. In some cases, tis is detectable simply looking at te velocity field (see [9]), wic features nonpysical oscillations. However, as we sall see in Section 6, in some cases te velocity field does not display oscillations, yet it does not correspond to te pysical solution (see, e.g., Fig. 11(a), were te velocity computed by DNS in not oscillating, but it grossly underestimates te measured jet lengt). A possible remedy to tis issue is to introduce a model wic filters te velocity and conveys te energy lost to resolved scales. Suc models can be tougt as a way to numerically revert te energy cascade by transferring te energy dissipated at te small (unresolved) scales towards larger (resolved) scales. 4

6 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 5 Te Leray model couples te Navier-Stokes equations wit a differential filter. Te model can be written as ρ t u + ρ (u )u 2µ s u + p = f in Ω (t 0, T ), (13) u = 0 in Ω (t 0, T ), (14) 2δ 2 (a(u) s u) + u + λ = u in Ω (t 0, T ), (15) u = 0 in Ω (t 0, T )., (16) Here, u is te filtered velocity, δ can be interpreted as te filtering radius (tat is, te radius of te neigborood were te filter extracts information from te unresolved scales), te variable λ is a Lagrange multiplier to enforce te incompressibility constraint for u and a( ) is a scalar function suc tat a(u) 0 were te velocity u does not need regularization a(u) 1 were te velocity u does need regularization. Tis function, usually referred to as indicator function, is crucial for te success of te Leray model. In te next section, we will discuss some of te possible coices for a( ) tat ave been proposed. Here, we mention tat te coice a(u) 1 corresponds to te classic Leray-α model [33]. Tis coice as te advantage of making te operator in te filter equations linear and constant in time, but its effectivity is rater limited, since it introduces te same amount of regularization in every region of te domain, ence causing overdiffusion. Equations (15)-(16) require suitable boundary conditions. Tese are cosen to be u = u D on Ω D (t 0, T ), (17) (2δ 2 a(u) s u λi)n = 0 on Ω N (t 0, T ), (18) wile no initial condition is required for u, since tere is no time derivative in eq. (15)-(16). Te impact of non-diriclet boundary conditions as not been investigated torougly in literature. In Section 4, we will discuss te effect of tese boundary conditions on te solution of te problem. Even toug (13)-(14) are linear in (u, p) and te filter problem is linear in (u, λ), te coupling is non-linear, due to te term (u )u in eq. (13), and a(u) s u in eq. (15) wen a( ) is not constant. Remark 2.2 Te structure of te filter problem can be qualified as a generalized Stokes problem wit a nonconstant viscosity. A solver for te filter can ten be obtained by adapting a standard linearized Navier-Stokes solver. Tis will be discussed in Section 5, were we will address preconditioning of te two saddle point problems Time discrete problem To discretize in time problem (13)-(16), let t R, t n = t 0 + n t, wit n = 0,..., N T and T = t 0 + N T t. Moreover, we denote by y n te approximation of a generic quantity y at te time t n. In te following we will denote by Ω te domain of te equations. For te time discretization of system (13)-(16), we adopt a Backward Differentiation Formula of order p (BDFp), see e.g. [39]. Te Leray system discretized in time reads: given u 0, for n 0 find te solution (u n+1, p n+1, u n+1, λ n+1 ) of te system: ρ α t un+1 + ρ u n+1 u n+1 2µ s u n+1 + p n+1 = b n+1, (19) u n+1 = 0, (20) 2δ 2 (a(u n+1 ) s u n+1) + u n+1 + λ n+1 = u n+1, (21) u n+1 = 0, (22) 5

7 6 L. BERTAGNA ET AL. were α is a coefficient tat depends on te order of BDF cosen, and b n+1 contains te forcing term f n+1 and te solution at te previous time steps used to approximate te time derivative of u at time t n+1. For example, wen using BDF2, we ave t u 3un+1 4u n + u n 1, (23) 2 t tus α = 3/2 and b n+1 = f n+1 + (4u n u n 1 )/(2 t). A monolitic approac for problem (19)-(22) would lead to ig computational costs, making te advantage compared to DNS questionable. To decouple te Navier-Stokes system (19)-(20) from te filter system (21)-(22) at te time t n+1, we ave two options: 1. Filter-ten-solve: Solve te filter equations (21)-(22) first, wit u n+1 replaced by a suitable extrapolation u and a(u n+1 ) replaced by a(u ), and ten solve equations (19)-(20) wit advection field given by te filtered velocity previously computed. 2. Solve-ten-filter: Solve equations (19)-(20) first, replacing te advection field u n+1 wit a suitable extrapolation u, and ten solve te filter problem (21)-(22). In eiter approac, one could iterate between te two subproblems, using a fixed point approac to te fulfillment of a certain convergence criterion. However, to keep te computational costs low, we adopt a semi-implicit approac, by performing only one iteration per time step. Approac 1 is investigated torougly in [9] using te linearly extrapolated Crank-Nicolson approximation u = (3u n u n 1 )/2. In tis paper we will focus on approac 2 wit a modified version of te EFR sceme proposed in [32]. Te EFR algoritm reads as follows: given te velocities u k (k = n p + 1,..., n) needed for te approximation of t u by BDFp at t n+1, i) evolve: find intermediate velocity and pressure (v n+1, q n+1 ) suc tat ρ α t vn+1 + ρ u v n+1 2µ s v n+1 + q n+1 = b n+1, (24) v n+1 = 0, (25) were u is a suitable approximation of te end-of-step velocity u n+1 based on previous timesteps solutions. ii) filter: find (v n+1, λ n+1 ) suc tat 2δ 2 (a(v n+1 ) s v n+1) + v n+1 + λ n+1 = v n+1 v n+1 = v n+1 = 0 iii) relax: set were χ [0, 1] is a relaxation parameter. u n+1 = (1 χ)v n+1 + χv n+1, (26) p n+1 = q n+1, In [32] te autors use energy arguments to support te coice χ = O( t), in order to keep te numerical dissipation low. Te EFR algoritm as te advantage of modularity: being te problems at steps i) and ii) numerically standard, tey can be solved wit legacy Navier-Stokes solver. It was sown in [36] tat te EFR algoritm is equivalent to a certain viscosity model in Large Eddy Simulation. 6

8 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 7 3. INDICATOR FUNCTION FOR NONLINEAR FILTERS: PHYSICS VS MATHEMATICS BASED Te breaking down of eddies into smaller ones until tey get damped is a igly nonlinear process, owever most current models use linear filters to select te eddies to be damped. Linear filter based stabilization, developed by Boyd [10] and Fiscer and Mullen [18, 35], as been widely studied over te past years (see, e.g., [34, 50, 21]). Nonlinear filters ave been considered in [32], were damping of eddies is based on nonlinearity of real flow problems. Te success of nonlinear filters suc as (13)-(16) in simulations ultimately depends on te reliability of te indicator function. One of te most matematically convenient indicator function is a(u) = u (suitably normalized [7]) because of its strong monotonicity properties. Wit tis coice, we recover a Smagorinsky-like model, wic is owever known to be not sufficiently selective. In fact, it selects laminar sear flow (were u is constant but large) as regions of te domain wit large turbulent fluctuations. In te following we report on some indicator functions tat ave been proposed in te literature. We group tem into two categories: pysical penomenology based and deconvolution based. We igligt teir strengts and limitations Pysical penomenology based indicator functions Te indicator functions proposed in [32] are based on pysical quantities tat are known to vanis for coerent flow structures. One of te most popular metods for eduction of coerent vortices is te Q criterion [29], wic identifies as persistent and coerent vortex tose regions were: Q(u, u) = 1 2 ( ss u : ss u s u : s u) > 0, were te spin tensor ss u and te deformation tensor s u were defined in (7) and (6), respectively. From te definition it is apparent tat Q > 0 occurs in tose regions were spin dominates deformation. An indicator is obtained by rescaling Q(u, u) so tat te condition Q(u, u) > 0 implies a(u) 0, tat is, regularization is not needed. Te indicator function based on te Q-criterion is ten given by a Q (u) = π arctan ( δ 1 Q(u, u) Q(u, u) + δ 2 A second indicator uses an eddy viscosity coefficient formula proposed by Vreman [51] tat vanises for 320 types of flow structures known to be coerent. Te Vreman based indicator function reads B(u) a V (u) = u 4, F were te subindex F refers to te Frobenius norm and B(u) is defined as B = β 11 β 22 β β 11 β 33 β β 22 β 33 β 2 23, β ij (u) = ). m=1,2,3 u i x m u j x m. Since 0 B(u)/ u 4 F 1, a V (u) [0, 1]. Te Vreman based indicator function was sown to be successful in [8]. Anoter pysics based indicator function uses te relative elicity density RH, wic is a local quantity, its macroscopic counterpart being te elicity H. Te two quantities H and RH are defined respectively as H = 1 u w dω, Ω Ω RH = u w u w, 7

9 8 L. BERTAGNA ET AL. wit w denoting vorticity, i.e. w = u. From te Navier-Stokes equations in rotational form, it is possible to see tat local ig elicity suppresses local turbulent dissipation caused by breakdown of eddies into smaller ones. Te elicity based indicator is developed by adjusting relative elicity density so tat values of RH near one imply a(u) 0. It reads a H (u) = 1 u w u w + δ 2. Notice tat oter (more selective) indicator functions can also be obtained by taking te geometric average of two (or more) indicator functions. Te indicator functions discussed in tis section ave te advantage of requiring only algebraic operations on u and its derivatives. Teir implementation may be quite straigtforward. However, te major drawback is tat tey do not allow for a rigorous convergence teory to verify te robustness of te associated filtering metod Deconvolution based indicator functions Let F be a linear, invertible, self-adjoint, compact operator from a Hilbert space V (suc as H 1 (Ω) or H 1 0 (Ω)) to itself. Te spectral teorem gives (see, for instance, [11]) F x = λ i x, e i e i, F 1 1 y = y, e i e i, λ i i=0 were λ i are te eigenvalues of F, and e i are te corresponding eigenfunctions, wic form an ortonormal basis for V. Since F is compact, te inverse operator F 1 (i.e. te operator s.t. x = F 1 F x) is unbounded. Let D be a bounded regularized approximation of F 1, wose action on y is given by wit Ten Dy = i=0 i=0 ( ) 1 φ y, e i e i λ i ( ) 1 1 if i is small, φ λ i λ i 0 if i is large. { small if x is smoot, x DF x is large if x is not smoot, were smoot is intended wit respect to te eigenfunction of te operator F. In particular, x is smoot if x, e i is significantly different from zero only for small values of i. Te composition of te two operators F and D can be interpreted as a low-pass filter. Tis motivates te indicator function 3.3. Te Van Cittert deconvolution operator a D (u) = u D(F (u)). (27) A popular coice for D is te Van Cittert deconvolution operator D N, defined as D N = N (I F ) n. n=0 Te evaluation of a D wit D = D N (deconvolution of order N) requires ten to apply te filter F a total of N + 1 times. If I F is spectrally bounded by 1, ten D N can be seen as te truncated 8

10 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 9 Neumann expansion of F 1, and D N approaces F 1 as N. However, tis does not introduce a limitation for te application of te metod. Since F 1 is not bounded, in practice N is cosen to be small, as te result of a trade-off between accuracy (for a regular solution) and filtering (for a non-regular one). In tis paper we consider N = 0, 1, corresponding to D 0 = I and D 1 = 2I F. For tese coices of N, te indicator function (27) becomes a D0 (u) = u F (u), a D1 (u) = u 2F (u) + F (F (u)). (28) Remark 3.1 In order to ensure tat a(u) [0, 1], indicator function (27) is rescaled a D (u) = u D(F (u)) max(1, u D(F (u)) ). (29) Notice tat te Van Cittert deconvolution D N can also be interpreted also as te N-t iteration of a Ricardson sceme to solve te problem F (u) = b. In fact, letting u N = D N (b), we ave N+1 N+1 u N+1 = (I F ) n (b) = b + (I F ) n (b) = n=0 = b + (I F ) n=1 N (I F ) n (b) = b + (I F )(u N ) = u N + (b F (u N )). (30) n=0 From tis perspective, we can furter support te coice of limiting N. In fact, since te operator F is compact, its eigenvalues accumulate to 0 and its inverse leads to an ill-posed problem. Among te possible regularization tecniques used to deal wit ill-posed problem, one is precisely to use iterative metods (suc as Ricardson) wit a limited number of iterations. Te Van Cittert deconvolution can be tus regarded as an iterative regularization of te inverse problem F (u) = b Te Van Cittert-Helmoltz operator We select F to be te linear Helmoltz filter operator F H [23] defined by F = F H ( I + δ 2 L ) 1, L = 3 2 x 2 i=1 j. It is possible to prove [15] tat φ D N (F H (φ)) = δ 2N+2 L N+1 F N+1 H φ = ( 1)N+1 δ 2N+2 N+1 F N+1 H φ. (31) Terefore, a DN (u) is close to zero in te regions of te domain were u is smoot. Indicator function (27) wit D = D N and F = F H as been recently proposed in [9], owever te idea of using van Cittert approximate deconvolution in fluid models to increase accuracy is well establised and matematically grounded [46, 47, 15]. Notice tat te Van Cittert-Helmoltz deconvolution operator D N can be conveniently interpreted in a different manner, in view of te teory of maximal monotone operators and teir Yosida regularized operator (see [11], C. 7). Following tis teory, F H is te resolvent J δ 2 of L. Correspondingly, te Yosida approximation (or regularization) of L reads L δ 2 δ 2 (I J δ 2) = δ 2 (I F H ) If we let a N (u) = u D N (F H (u)), so tat a DN (u) = a N (u), from te recurrence equation (30) it follows tat a N (u) = (I F H )a N 1 (u) =... = (I F H ) N a 0 (u) = δ 2N+2 L N+1 δ u 2 9

11 10 L. BERTAGNA ET AL. or, in terms of te (scalar) indicator function a DN (u), a DN (u) = δ 2N+2 L N+1 δ 2 u. (32) Here, we list some properties we infer from Proposition 7.2 in reference [11]. Let v be a generic function in H 1 (Ω). Ten, 1. L δ 2v = LJ δ 2v 2. (L δ 2v, v) 0 3. lim δ 2 0 J δ 2v = v 4. δ 2 L δ 2v v. Notice tat (31) follows promptly from 1. In addition, if v H 2 (Ω) we also ave 5. L δ 2v = J δ 2Lv 6. L δ 2v Lv 7. lim L δ2v = Lv δ 2 0 ( lim δ 2 0 δ2 L δ 2v = 0). Assuming enoug regularity for v, by induction it is possible to prove tat te statement in brackets at point 7 olds for any N > 0, more precisely a DN (u) = δ 2N+2 L N+1 δ (u) vanises as te filter 2 radius approaces zero. 4. NONLINEAR FILTERING AS AN OPERATOR SPLITTING METHOD In tis Section we consider te EFR sceme introduced in Section 2.3, wit te indicator function defined in (29), D = D N and F = F H, and we propose an operator splitting interpretation. In Section 4.1, we investigate ow filtering affects te boundary conditions actually fulfilled by u n+1 and p n+1, wit particular attention to Neumann conditions. In Section 4.2 we propose a euristic pysical argument to coose te relaxation parameter χ. For te space discretization we use te Finite Element metod. For a given mes, we introduce te discretization parameter, wic we take to be te lengt of te sortest edge in te mes. In particular, we use an inf-sup stable Finite Element pair (e.g. Taylor-Hood elements [42]). More details concerning space discretization will be covered in section 5. We denote by f te discrete FE approximation of a generic continuous function f accordingly. To rewrite te EFR sceme as an operator splitting metod, let us define u a suitable approximation of te end-of-step velocity u n+1 extrapolated from previous time-steps. Let us ten denote by (v n+1, q n+1 ) te solution to te filter step. In addition we set µ = ρ δ2 t a D N (v n+1 ) = ρ δ2n+4 L N+1 t δ v n+1 2. (33) Here, te parameter δ is cosen to be of te order of. Notice tat µ is dimensionally a dynamic viscosity, a fact tat we will exploit in Section 4.2 to establis a practical rule for te selection of te relaxation parameter χ. Finally, we introduce te operators L NS [u ]v = ρ(u )v (2µ s v ),, (34) L F [v ]v = (2µ (v ) s v ). (35) Here, te notation A[v]u means tat te operator A is computed at v and ten applied to te function u. Notice tat te operator L F depends on v troug te artificial viscosity µ in (33). 10

12 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 11 Let (v n+1, q n+1 ) be te solution of te evolve step, tat reads We write te filter step as ρ α t vn+1 + L NS [u ]v + q n+1 = b n+1, (36) ρ vn+1 t Te relaxation step for te velocity reads v n+1 = 0, (37) v n+1 = u n+1 D on Ω D, (38) σ(v n+1, q n+1 ) n = g n+1 on Ω N. (39) + L F [v ]v + q n+1 Let us multiply (40) by χα and add it to (36) to obtain = ρ vn+1 t (40) v n+1 = 0 (41) v n+1 = u n+1 D on Ω D, (42) (2µ s v n+1 q n+1 I)n = 0 on Ω N, (43) u n+1 = (1 χ)v n+1 + χv n+1. (44) ρ α ((1 χ)vn+1 + χv n+1 ) + L t NS [u ]v n+1 + χαl F [v n+1 ]v n+1 + (q + χαq ) = b n+1. Ten, using te relaxation for te velocity (44), we obtain tat u n+1 satisfies ρ α t un+1 + L NS [u ]v n+1 + χαl F [v n+1 ]v n+1 + (q + χαq ) = b n+1. (46) Using one more time (44), we obtain (45) ρ α t un+1 + L NS [u ]u n+1 + (q n+1 + αχq n+1 )+ (47) χ(l NS [u ](v n+1 ) + αl F [v n+1 ]v n+1 ) = b n+1. Te continuity equation for u n+1 is automatically satisfied, since bot v n+1 and v n+1 are bot divergence free. Equation (47) suggests a relaxation for te end-of-step pressure, namely Te operator-splitting sceme involves tus tree steps: p n+1 = q n+1 + αχq n+1. (48) i) solve equations (36)-(37) to obtain te intermediate velocity and pressure using te standard Navier-Stokes operator L NS wit te pysical viscosity µ; ii) solve equations (40)-(41) to obtain te filtered velocity and pressure using te filter operator L F wit te artificial viscosity µ ; iii) finally, we combine te solutions found at steps i) and ii) wit (44) and (48) to get te end-ofstep velocity and pressure. In te tird step, te velocity and pressure found at te first step are corrected by taking into account te energy dissipated at te scales tat were not resolved wit te given mes in te Navier- Stokes step. Notice tat equation (47) is a consistent perturbation of te original Navier-Stokes 11

13 12 L. BERTAGNA ET AL. problem, te perturbation vanising wit coefficient χ tat in turn vanises wit te discretization parameters (recall tat, as suggested in [32], χ = O( t)). Notice in particular tat for δ 2 / t 0 we ave tat µ 0 and v n+1 v n+1. Recalling tat δ, tis means tat for a practical coice t te perturbed equation (47) is consistent wit te Navier-Stokes equations. On te oter and, wen falls below te Kolomogorv scale η, all te relevant scales are solved and no filter is needed. Te diffusive effects of te EFR sequence may be stated in te following Proposition. Proposition 4.1 Assume tat we ave omogeneous Diriclet boundary conditions. If χ [0, 1], at eac time step te following inequality olds u n+1 L 2 v n+1 L 2. (49) Proof Multiply (40) by v n+1 and integrate by parts. Since µ is positive and finite, we ave tat v n+1 2 L + 2 C vn+1 2 L 2 vn+1 2 L 2, were C depends on µ ρ, α and t. Ten, we rewrite te relaxation step as Notice tat v n+1 = u n+1 + χ(v n+1 ). (v n+1, u n+1 ) = (v n+1, (1 χ)v n+1 + χv n+1 ) = (v n+1, (1 χ)v n+1 (1 χ)v n+1 + v n+1 ) = (1 χ) v n+1 2 L + 2 (vn+1, v n+1 ) = (1 χ) v n+1 2 L + 2 C vn+1 2 L 2 For χ [0, 1] (see Remark 6.1), te last line is for sure positive. We ave v n+1 2 L = 2 (vn+1, v n+1 ) = (u n+1 + χ(v n+1 ), u n+1 + χ(v n+1 )) = u n+1 u n+1 2 L 2. 2 L + 2 2χ(vn+1, u n+1 ) + χ 2 v n+1 2 L 2 Tis proposition sows tat te filter actually damps te intermediate velocity in te L 2 norm. We speculate tat a similar relation formally olds for N+1 u n+1 and N+1 v n+1. Tis will be investigated in te follow-up of te present work. We rewrite (47) as ρ α t un+1 + L NS [u ]u n+1 + ( q n+1 + αχq n+1 ) + (50) χ ( L NS [u ](v n+1 ) + αl F [u n+1 ]v n+1 + α ( L F [v n+1 ] L F [u n+1 ] ) v n+1 ) = b n+1. Sould our speculation be true, te last term at te left and side is positive. If we now consider te operator L defined by L [u, u n+1 ]u n+1 = L NS [u ]u n+1 + αχl F [u n+1 ]u n+1. (51) and te perturbed discrete Navier-Stokes momentum equation ρ α t un+1 + L [u, u n+1 ]u n+1 + p n+1 = b n+1, (52) our arguments point out tat te EFR algoritm is an inexact operator splitting of (52). On te oter and, (52) is te original Navier-Stokes momentum equation plus a viscous term surrogating at te mes scale te effect of energy dissipation at te unresolved scales. 12

14 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 13 Remark 4.1 Te coice of te end-of-step pressure (48) is someow arbitrary. It is driven by te strategy of gatering all te effects of te filter to te velocity. Oter coices are possible. For instance, one may argue tat velocity and pressure sould be relaxed in te same way, leading to p n+1 = (1 χ)q n+1 + χq n+1, (53) Tis approac - tat we call relaxation-consistent - introduces a different perturbed equation for te EFR algoritm, i.e. ρ α t un+1 + L NS [u ]u n+1 + p n+1 + χ(l NS [u ](v n+1 ) + α(1 χ)l F [v n+1 ](v n+1 )) + χ (q n+1 + (α 1)q n+1 ) = b n+1. (54) Consistency is still preserved wit te same factor O(χ). Te accuracy and stability of tis sceme need to be furter investigated (see Sect. 5.1). Tus, te results in te Sect. 6 refer to (48) Boundary conditions for te end-of-step solution To conclude our description of te EFR algoritm, we address te boundary conditions satisfied by te end-of-step solution, so to justify te coice of boundary condition (43). By combining (44), (38), and (42), we ave u n+1 = u n+1 D, on Ω D. As for te Neumann part of te boundary Ω N, from (44) and (48), we ave (2µ s u n+1 p n+1 I)n = (2µ s ((1 χ)v n+1 + χv n+1 ) (q n+1 + αχq n+1 )I)n = (2µ s v n+1 q n+1 I)n + χ(2µ s (v n+1 ) αq n+1 I)n. Ten, using (39) and (43), we find tat (2µ s u n+1 p n+1 I)n = g n+1 χ(2µ s (v n+1 ) + 2αµ s v n+1 )n. (55) Tis sows tat wit te coice (43) for te filter, te end-of-step velocity and pressure fulfill a perturbation of te original Neumann conditions, yet consistent wit te coefficient χ. Remark 4.2 In te case of te more general Robin boundary condition of te form (2µ s u n+1 p n+1 I)n + γu n+1 = g n+1 on Ω R, (were in general γ > 0), te same consistency as for (55) can still be acieved by imposing te following boundary condition for te filter (2µ s v n+1 q n+1 I)n + γ α vn+1 = γ α vn+1 on Ω R. Tese boundary conditions can arise in particular applications, suc as, for instance, domain decomposition tecniques in geometric multiscale approaces (see, e.g., [19], Capter 11), or wit te use of te so called transpiration tecniques for Fluid-Structure Interaction problems (see, e.g., [14]) Quantification of te relaxation parameter χ As we noticed, for δ, δ 2 / t 0, χ t, te end-of-step pair (u n+1, p n+1 ) fulfills a consistent perturbation of te Navier-Stokes momentum equation (te mass conservation being fulfilled 13

15 14 L. BERTAGNA ET AL. exactly). However, we need a practical way to tune te proportionality constant c in te rule χ = c t. We experienced tat te coice c 1 does not provide enoug numerical dissipation in realistic applications (see Section 6). On te oter and, we postulate tat χ > 0 only for > η, since for 0 we do not need any filtering. In order to find a proper formula for χ we use a euristic argument. We set χ suc tat te viscous stress in (52) on an under-resolved mes of size provides te same amount of dissipation as te viscous term in (36) on a fully resolved mes of size η. Let us introduce an equivalent stress tensor in te perturbed Navier-Stokes equations (52) σ n+1 = p n+1 I + 2(µ + αχµ ) s u n+1. In addition, let s ξ denote te symmetric gradient on a mes wit size ξ, were ξ is eiter or η. We require tat te viscous contribution of tis equivalent tensor matces te viscous contribution of te standard stress tensor of a Newtonian fluid (5) on a mes of size η, Wit te approximation s ξ wit ξ 1, we obtain (µ + αχµ ) s u n+1 µ s ηu n+1, (56) (µ + αχµ ) 1 µ 1 η, leading to χ µ ( ) αµ η 1 µ 1 αρ a δ 2 ( ) η 1 t. (57) Here, a is te infinity norm of ( te indicator ) function. As for Remark 3.1, in practice a 1 so tat te constant c = µ 1 αρ a δ 2 η 1 is promptly estimated a priori. In practice we may set c = µ max( η, 0). αρηδ2 Remark 4.3 Te coice of te filtering radius δ is still an open problem wen dealing wit non-uniform grids. As we mentioned before, it is a common coice to set δ equal to te space discretization parameter, wic usually refers to te largest diameter of te elements of te mes. However, wen using non-uniform grids, tis could lead to eccessive numerical diffusion, since te region were te filter as a significant effect would not be guaranteed to be confined witin a single element and may include also one (or more) neigboring elements for te smallest elements. Terefore, in tis paper we set to be te lengt min of te sortest edge in te mes, wic guarantees tat for eac degree of freedom (d.o.f.) te region were te filter as a significant effect is confined witin te patc of elements saring te d.o.f. 5. DISCRETIZATION OF THE OPERATOR SPLITTING ALGORITHM Let T be a conformal and quasi-uniform partition of Ω. For te approximation of velocity and pressure, we use inf-sup stable FE Taylor-Hood pair P 2 -P 1 [42, 16]. We do not use any specific numerical stabilization for te convective term. For te time discretization we use BDF2 (23), wit te corresponding convective term extrapolation u = 2u n u n 1. We denote by M te mass matrix, K te diffusion matrix, N te matrix associated wit te discretization of te convective term, and B te matrix associated wit te discretization of te 14

16 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 15 operator ( ). Te full discretization of problem (36)-(39) wit BDF2 yields te following system ρ 3 2 t Mvn+1 + ρnv n+1 + µkv n+1 + B T q n+1 = b n+1 u, (58) Bv n+1 = 0, (59) were v n+1 and q n+1 are te arrays of nodal values for te intermediate velocity and pressure. Te array b n+1 u accounts for te contributions of solution at te previous time steps and te contribution coming from nonomogeneous boundary conditions, wile te constant 3/2 in te first term comes from (23). Remark 5.1 In case of nonomogeneous Diriclet boundary conditions, depending on te implementation details, te rigt and side of (59) may also be nonzero. Tis can be te case, for instance, wen a lift function is used and te degrees of freedom on te Diriclet boundary are eliminated (via Gaussian reduction). In our implementation, owever, te Diriclet boundary conditions are imposed by setting te corresponding rows in te momentum equation to be u i = u D (x i ). For tis reason, te rigt and side of (59) is still 0. Setting C = ρ 3 2 tm + ρn + µk, we can rewrite (58)-(59) in te form Ax n+1 = b n+1, (60) were [ C B T A = B 0 ] [ ] [ ], x n+1 v n+1 = q n+1, b n+1 b n+1 u =. (61) 0 Let K be te matrix associated wit te discretization of te diffusive term in (40). Te full discretization of problem (40)-(43) ten yields ρ t Mvn+1 + Kv n+1 + B T q n+1 = ρ t Mvn+1, (62) Bv n+1 = 0, (63) were v n+1 and q n+1 are te nodal values of te filter step velocity and pressure. Setting C = ρ tm + K, we can rewrite (62)-(63) in te form Ax n+1 = b n+1, (64) were [ C B T A = B 0 ] [, x n+1 v n+1 = q n+1 ] [ ρ, b n+1 = t Mvn+1 0 ]. (65) At every time level t n+1, to solve systems (60) and (64) we use te left preconditioned GMRES metod. To precondition bot systems, we use an upper-triangular variant of te pressure corrected Yosida splitting [22, 44]. For te matrix A, tis preconditioner reads: [ C B T P A = 0 S(S + BH(µK + ρn)hb T ) 1 S ], H = 2 t 3ρ M 1, S = BHB T. (66) Te above preconditioner is a suitable approximation of te U factor in te exact block LU factorization of matrix A in (61): [ A = LU, L = I 0 BC 1 I ] [ C B T, U = 0 BC 1 B T 15 ]. (67)

17 16 L. BERTAGNA ET AL. See [40, 41, 24] for more details. For te matrix A, te preconditioner as a similar structure, namely: [ ] C B T P A = 0 S(S + BH(K)HB T ) 1, H = t S ρ M 1, S = BHB T. (68) Te application of te preconditioner requires to solve two linear systems in bot C and S for P A, and two linear systems in bot C and S for P A. To solve eac of tese systems, we use a Krylov metod wit a general purpose preconditioner, suc as incomplete LU or algebraic multilevel. It is wort mentioning tat, wile C is in general non-symmetric because of te convective term, te matrix C is symmetric (and positive definite). Terefore, wile for P A te (1,1) block is solved wit GMRES metod, (1,1) block of P A can be solved wit te Conjugate Gradient metod Matrix formulation of te relaxation consistent EFR sceme As done for te matrix A we can perform te LU factorization of te matrix A, [ ] [ ] I 0 C B T A = LU, L = BC 1, U = I 0 BC 1 B T. (69) Wit tis notation, te relaxation consistent EFR metod mentioned in Remark 4.1 can be promptly rewritten in a compact form. According to tis sceme we set at eac time step [ ] [ ] [ ] u n+1 v n+1 v n+1 = (1 χ) + χ, p n+1 q n+1 q n+1 were [ u n+1 p n+1] T is te end-of-step solution. By getting rid of te filter quantities [ v n+1 q n+1] T, we obtain [ ] [ ] ( [ u n+1 v n+1 ρ ]) [ p n+1 = q n+1 χ I (LU) 1 t M 0 ] v n+1 B 0 q n+1. By straigtforward (tedious) computations, we can rewrite te matrix in parentesis (multiplied by χ) as [ ( I C 1 B T Σ B) 1 (I ) ] (I + Z) 1 0 Σ 1 B ( I (I + Z) 1) I were Σ = BC 1 B T and Z = t ρ M 1 K. Tis representation of te solution at eac time step outlines ow te end-of-step solution is te result of a modification of te unfiltered one [ v n+1 q n+1] T weigted by te relaxation parameter χ. Notice in addition tat ( I (I + Z) 1 ) = Z(I + Z) 1 = t ρ M 1 K(I + Z) 1 O( t). Tis means tat te perturbation induced by tis sceme on te velocity scales wit χ t (as opposed to χ only ). Specific strategies to manage te correct amount of filtering for tis relaxation consistent sceme will be investigated in a fortcoming work. 6. NUMERICAL RESULTS In order to demonstrate te effectiveness of te approac described in te previous sections, we ave selected a bencmark from te U.S. Food and Drug Administration (FDA). Tis bencmark consists in simulating te flow of an incompressible and Newtonian fluid wit prescribed density 16

18 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 17 Figure 1. A section of te computational domain, wit D i = 0.012, D t = 0.004, L i = 4D i and L o = 12D i. Te units are meter. Re t Re i flow rate Q (m 3 /s) η (m) e e e e-5 Table I. Troat Reynolds number Re t, inlet Reynolds number Re i, flow rate Q, and Kolmogorov lengt scale η for te flow regimes under consideration. and viscosity (ρ = 1056 kg/m 3 and µ = Pa s) in an idealized medical device saped like a nozzle (see Fig. 1) at different Reynolds numbers. In te geometry pictured in Fig. 1, te fluid flows from left to rigt, passing troug a cylindrical entrance region, a conical convergent, a cylindrical troat, and a sudden expansion into a larger cylinder. Te complete FDA bencmark requires to study tis system for a variety of conditions, including laminar, transitional, and turbulent regimes: te results of te publised inter-laboratory experiments refer to values of te Reynolds numbers in te troat (defined as in (10)) of Re t = 500, 2000, 3500, 5000, In a previous work [37], we ave successfully validated LifeV [3] against tis bencmark for Re t up to 3500 using DNS. To test our metodology, we focus on Reynolds numbers Re t = 3500, Te case Re t = 6, 500 is carefully analyzed in anoter paper [6]. In Table I, we report te troat Reynolds number Re t, te corresponding inlet Reynolds number Re i, flow rate Q, and te Kolmogorov lengt scale η for te considered flow regimes. Te value of η was found by plugging into (12) te value of Re t and te diameter of te expansion cannel D i as caracteristic lengt. Notice tat for bot flow regimes te flow in te entrance region is laminar, Re i being below te critical Reynolds number for transitional flow in a straigt pipe (Re 2000 [43]). On te lateral surface of te computational domain we prescribe a no-slip boundary condition. For te two flow regimes in Table I, at te inlet section we prescribe a Poiseuille velocity profile to get te desired flow rate, a coice wic is justified by te considered values of Re i. Te lengt of te inlet camber L i was set to four times its diameter. At te outlet section, we prescribe a stressfree (natural) boundary condition. Te actual experimental set up of te FDA bencmark is a closed loop [27] and it is not reflected by our condition. However, omogeneous Neumann conditions are expected to introduce a minimal error localized only in te close neigborood of te outlet section [28]. Te results of te flow analysis are not affected, provided tat te expansion cannel is long enoug. For all te simulations, te lengt of te expansion cannel (L o in Fig. 1) was set to 12 times its diameter and we cecked tat in all te cases te velocity components reac a plateau before te outlet. As for te initial condition, we start our simulations wit fluid at rest, i.e., u = 0 everywere in Ω. We use a smoot transition of te velocity profile at te inlet from rest to te regime flow conditions. For bot flow regimes in Table I, we considered several meses wit different levels of refinement. Te selection of te time step was driven by accuracy considerations solely. In fact, even toug te semi-implicit treatment of te convective term in eq. (36) does not guarantee te unconditional stability in time, we encounter no stability issues in te numerical experiments. 17

19 18 L. BERTAGNA ET AL. mes name min avg max # nodes # tetraedra t 1900k 1.06e e e-3 3.7e5 1.9e6 1e k 1.08e e e-3 2.3e5 1.2e6 1e-4 900k 1.09e e e-3 1.8e5 9e5 1e-4 330k 2.23e e e-3 6.5e4 3.3e5 2e-4 140k 3.39e e e-3 3.1e4 1.4e5 3e-4 Table II. Case Re t = 3500: meses used for te simulations, wit teir minimum diameter min, average diameter avg, maximum diameter max, and number of nodes and tetraedra. Te units for te diameters are meter. We also report te time step t (in s) used for te simulations wit eac mes. We compare te experimental data provided by te FDA wit our numerical simulations for te flow regimes listed in Table I. Te experimental data were acquired by tree independent laboratories and one of te laboratory ran tree trials, so tat for eac case we ave five sets of data. Te comparison is made in terms of normalized axial component of te velocity and normalized pressure difference along te centerline. Te axial component of te velocity u z is normalized wit respect to te average axial velocity at te inlet ū i u norm z = u z, wit ū i = Q (70) ū i /4, were Q is te volumetric flow rate calculated from te troat Reynolds number (see Table I). Te pressure difference data are normalized wit respect to te dynamic pressure in te troat πd 2 i p norm = p z p z=0 1/2ρū 2, wit ū t = Q t πdt 2 /4, (71) were p z denotes te wall pressure along te z axis and p z=0 is te wall pressure at z = 0. To compute te value of p norm, we probed te pressure value at te corresponding location on te axis of te domain, since we observed pressure values being approximately uniform on axial crosssections. Te graps wit te above comparisons are reported in Sec. 6.1 and 6.2 for Re t = 3500 and Re t = 5000, respectively Case Re t = 3500 Among te Reynolds numbers considered by te FDA bencmark, Re t = 3500 is te lowest above te critical Reynolds number for transitional flow in a straigt pipe. As mentioned earlier, we ave dealt wit tis case in [37], were we sowed tat DNS wit a properly refined mes is able to capture wit precision te jet breakdown observed in te experiments. Here, we start from te mes used in [37] for te DNS at Re t = 3500 and make it progressively coarser to understand te performances of te EFR algoritm described in Sec. 4. Te meses we considered and te associated time step used in te simulations are reported in Table II. Te name of eac mes reflects te number of elements. After several numerical experiments, in [37] we managed to identify a time step value and a mes sufficiently refined in te different regions of te domain suc tat te results obtained wit DNS were in excellent agreement wit te experimental data. Mes 1900k features tat level of refinement and is associated wit te same t used for te simulations in [37]. Te time step associated wit all te oter meses in Table II was cosen suc tat te ratio min / t is kept approximatively constant. Starting from fluid at rest, te turbulent regime is fully developed already at t 0.3 s. We let te simulations run till past tat time and ten take around 10 solutions wose average is compared to te experimental data. In fact, since te measurements of a turbulent flow are averaged over time [27], we average also te numerical results for a fair comparison. We noticed tat averaging over more tan 10 solutions does not cange te average value. A DNS is possible only wit meses 1900k and 1200k. In fact, a DNS wit mes 900k does not reac regime conditions for te instabilities in te computed velocity due to mes under-resolution 18

20 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 19 (a) normalized axial velocity along z (b) normalized pressure difference along z Figure 2. Case Re t = 3500, DNS wit two different meses: comparison between experimental data (solid lines) and numerical results (dased lines) for (a) normalized axial velocity (70) along te z axis and (b) normalized pressure difference (71) along te z axis. Te legend in (b) is common to bot subfigures. causing te simulation to cras. We report te comparison for te normalized axial velocity (70) along te z axis (Fig. 2(a)) and te normalized pressure drop (71) along te z axis (Fig. 2(b)). In Fig. 2, we plotted a dot for every measurement and a solid line to linearly interpolate te five sets of measurements, wile we used a dased line for te numerical results obtained wit meses 1900k and 1200k. We notice tat te axial velocities computed wit bot meses matc te measurements all along te portion of te z axis under consideration ( < z < 0.08), capturing accurately te jet breakdown point observed in experiments. As sown in Fig. 2(b), also te simulated pressure drop on bot meses is in very good agreement wit te experimental data, except in te convergent were te simulated pressure difference overestimates almost all te measurements. Te reason for tis overestimation is explained in [37]. Fig. 2 sows tat even mes 1200k as a sufficient level of refinement to obtain numerical results in excellent agreement wit te measurements in terms of average quantities, despite its avg is rougly 20 times larger tan te Kolmogorov lengt scale at Re t = 3500 (see Tables I and II). We test te EFR algoritm described in Sections 4 and 5 wit deconvolution of order N = 0 on all te meses in Table II coarser tan mes 1200k. We report te comparison between computed and measured normalized axial velocity (70) and pressure difference (71) in Fig. 3(a) and 3(b), respectively. Remarkably, te EFR algoritm succeeds in curing te convective term instabilities even on mes 140k, wic as 88% less elements tan mes 1200k (te coarsest mes tat allowed for a DNS) and an average diameter more tan 42 times larger tan η at Re t = From Fig. 3(a), we see tat te axial velocity computed on meses 900k and 330k are in agreement wit te measurements all along te z axis. Wit mes 140k, te jet starts to break down closer to te sudden expansion tan te jets obtained wit te oter two meses. Noneteless, te total jet lengt computed wit mes 140k agrees very well wit te measured total jet lengt. As for te pressure drop, Fig. 3(b) sows tat te computations on te tree meses fall witin te measurements. In particular, te pressure drop simulated wit mes 900k is in very good agreement wit te experimental data, again wit te exception of te convergent region. In [45], none of te presented CFD results was able to reproduce te correct jet breakdown point, because DNS predicted a longer jet (likely due to an incorrect simulation setup) wile simulations wit turbulence models under-predicted te jet lengt. Tus, we may state tat all te computed axial velocities in Fig. 3(a) are reasonable, regardless of te mes. For a qualitative comparison, we report in Fig. 4 te velocity magnitude computed wit meses 1200k, 330k, and 140k on a section of te domain after te turbulent regime is fully establised. Again, te results wit mes 1200k (in Fig. 4(a)) ave been obtained wit DNS and terefore sow 19

21 20 L. BERTAGNA ET AL. (a) normalized axial velocity along z (b) normalized pressure difference along z Figure 3. Case Re t = 3500, EFR wit tree different meses, N = 0: comparison between experimental data (solid lines) and numerical results (dased lines) for (a) normalized axial velocity (70) along te z axis and (b) normalized pressure difference (71) along te z axis. Te legend in (b) is common to bot subfigures. (a) mes 1200k (b) mes 330k (c) mes 140k Figure 4. Case Re t = 3500: velocity magnitude computed wit meses (a) 1200k, (b) 330k, and (c) 140k on a section of te domain after te turbulent regime is fully establised. Te results wit mes 1200k ave been obtained wit DNS, wile te results wit meses 330k and 140k ave been obtained wit te EFR algoritm and N = 0. a ig level of detail. Wit meses 330k and 140k, te finer details of te smaller turbulent structures are lost, yet tanks to te EFR algoritm te average beavior of te flow is well captured (see Fig. 4(b) and 4(c)) at a fraction of te computational cost. In fact, a time step of te DNS wit mes 1200k takes around 240 s on 80 CPUs, wile a time step of te EFR algoritm wit mes 330k takes around 80 s (50 s for te evolve step plus 30 s for te filter step) on 48 CPUs and wit mes 140k around 65 s (50 s for te evolve step plus 15 s for te filter step) on 24 CPUs. Tese computational times refer to simulations run on Maxwell, a cluster of te Researc Computing Center at te University of Houston. Next, we set te deconvolution order N to 1 and repeat te simulations on meses 900k, 330k, and 140k. We report te comparison for te normalized axial velocity (70) along te z axis in Fig. 5(a) and te normalized pressure difference (71) along te z axis in Fig. 5(b). From Fig. 5(a), we see tat, wile te axial velocity computed on mes 900k is still in excellent agreement wit te experimental data, te jet obtained wit te coarser meses is too long. At Re t = 3500 te coice N = 1 is more diffusive tan te corresponding case N = 0 wit te same relaxation parameter and on te same mes. We speculate tat tis is related to (32) and to Property 2 listed at te end of Sect In fact, if we assume tat te velocity is regular enoug so tat a DN ( ) is bounded independently of δ (as stated in te Property mentioned above), te amount 20

22 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 21 (a) normalized axial velocity along z (b) normalized pressure difference along z Figure 5. Case Re t = 3500, EFR wit tree different meses, N = 1: comparison between experimental data (solid lines) and numerical results (dased lines) for (a) normalized axial velocity (70) along te z axis and (b) normalized pressure difference (71) along te z axis. Te legend in (b) is common to bot subfigures. (a) N = 0 (b) N = 1 Figure 6. Case Re t = 3500, EFR wit mes 140k: value of χ over time interval [0.43, 0.68] s for (a) N = 0 and (b) N = 1. of viscosity introduced by our EFR sceme is proportional to δ 2 for N = 0 and to δ 4 for N = 1. For δ tis implies tat coarsening te mes of a factor of 2 increases te extra-diffusion of a factor of 4 for N = 0 and of a factor of 16 for N = 1. More in general, te beavior of te artificial viscosity (33) as N increases is te result of a complex interplay of te effect of te Yosida regularization (see Sec ) and te fact tat as N operator D N approaces F 1 (an unbounded operator, see Sec. 3.3). Tis will be investigated in future works. In Fig. 5(b), we see tat te pressure drop is progressively more overestimated in te entrance region as te mes gets coarser (up to 25% overestimation on mes 140k wit respect to te average measured pressure difference). Tis is reflected by te te value of χ over te time interval [0.43, 0.68] s for te mes 140k. As a matter of fact, te parameter χ selected following (57) canges in time because of variations of a. In Fig. 6(a), we see tat for N = 0 te value of χ over interval [0.43, 0.68] s takes constantly te value 0.1, wic corresponds to a = 1. Wen N = 1, te value of χ oscillates between 0.1 and 0.17 (see Fig. 6(b)). A larger value of χ corresponds to a more regular velocity detected by te indicator function tat gets smaller at te denominator of (57). Tis is reflected by te longer jets in Fig. 5 tat we get wit meses 330k and 140k for N = 1. 21

23 22 L. BERTAGNA ET AL. (a) normalized axial velocity along z (b) normalized pressure difference along z Figure 7. Case Re t = 3500, EFR wit mes 140k and four different values of te deconvolution order N = 0, 1, 2, 3: comparison between experimental data (solid lines) and numerical results (dased lines) for (a) normalized axial velocity (70) along te z axis and (b) normalized pressure drop (71) along te z axis. Te legend in (b) is common to bot figures. Notice tat te values of χ computed wit eq. (57) and sown in Fig. 6 are two orders of magnitude larger tan t used for mes 140k (see Table II). For te problem under consideration, te coice χ t is not appropriate, since it would lead to an under-diffused flow tat would not matc te experimental data. Remark 6.1 We see from Fig. 6 tat te value of χ for mes 140k is around 0.1 for N = 0 and less tat 0.2 for N = 1. In te proof of Proposition 4.1, we assumed χ [0, 1]. On one and, χ defined in (57) is clearly positive. In fact, it would be negative wen < η, wic means tat te mes is refined enoug for DNS and tere is no need for te filtering step. On te oter and, to ave χ > 1 for mes 140k (te mes wit te largest ratio /η among tose used at Re t = 3500) one would ave to take a time step 10 times larger tan te one we used. Suc a large time step would be inadequate to follow te pysics of te problem. Anoter possibility to ave χ > 1 would be to set δ = min / 10, wic is not appropriate as explained in Remark 4.3 As previously pointed out, te dependence of te solution and te overall smooting effects on N is not completely clear. In fact, te jet lengt reduces wen we compare te case N = 1 to N = 3, wile te cases N = 0 and N = 2 produces similar results - see Fig. 7(a) and 7(b). As for te pressure drop, te only outlier is te case N = 1, wile for N = 0, 2, 3 we get similar results. Tis non-trivial sensitivity to N deserves deeper investigations and it is subject of an ongoing work [6]. On te oter and, for N = 0 and a coarse mes wit elements and avg = m te simulation crases less tan 0.1 s after reacing regime conditions regardless of te deconvolution order. In tis case, overdiffusion is not enoug for stabilization purposes. A key role in te EFR algoritm is played by te indicator function. We sow in Fig. 8 te indicator function a D0 (see (28) and Remark 3.1 for te definition), computed wit meses 330k, and 140k at te same time step as te velocity magnitudes reported in Fig. 4(b) and 4(c), respectively. Te color bars refer to te value of te indicator function on a particular section of te domain, and not on te wole domain. For bot cases in Fig. 8, te indicator function takes its largest value in te boundary layer at te entrance of te troat. Moreover, on mes 140k it takes fairly large values all along te jet, wile on mes 330k larger values are taken only were te jet breaks down. Fig. 8 sows tat a D0 is a suitable indicator function since it correctly selects te regions of te domain were te velocity does need regularization. Remark 6.2 Wen a is very small, te value of χ becomes large. In order to avoid dealing wit a large 22

24 NONLINEAR FILTERING FOR INCOMPRESSIBLE FLOWS 23 (a) mes 330k (b) mes 140k Figure 8. Case Re t = 3500, EFR wit N = 0: indicator function a D0 computed wit meses (a) 330k and (b) 140k at te same time step as te velocity magnitudes reported in Fig. 4(b) and 4(c), respectively. (a) kinetic energy (b) power spectral density Figure 9. Re = 3500, DNS wit mes 1900k: (a) kinetic energy over time interval [0.29, 0.64] s and its (b) power spectral density. value of χ, in our solver te filter is turned on only wen te velocity is sufficiently large to make te current Kolmogorov lengt scale - computed wit te current Reynolds number - smaller tan min. Finally, let us analyze te evolution of te kinetic energy of te system. We let te DNS wit mes 1900k run for several tents of seconds. Te computed kinetic energy is sown in Fig. 9(a): it oscillates around te mean value Te periodic caracter of te solution can be figured out by means of a Fourier analysis of te kinetic energy in time (see, e.g., [4]). Te power spectral density of te signal in Fig. 9(a) is reported in Fig. 9(b). Significant frequencies reac up to 50 Hz. In Fig. 10, we see te power spectral density of te kinetic energy computed wit te EFR algoritm on meses 900k and 140k. Te comparison pinpoints tat wen we resort to coarser meses te amplitude damping at different frequencies is larger. However, te frequency of 50 Hz does not significantly cange also wit te coarsest meses. We conclude tat te EFR does not affect te dispersion - see Fig. 9(b). 23