A POD-Galerkin reduced order model for a LES filtering approach
AA POD-Galerkin reduced order model for a LES filteringapproach
Michele Girfoglio ∗ , Annalisa Quaini † , and Gianluigi Rozza ‡ SISSA, International School for Advanced Studies, Mathematics Area, mathLab, viaBonomea, Trieste 265 34136, Italy Department of Mathematics, University of Houston, Houston TX 77204, USA
Abstract
We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Or-der Model (ROM) for a Leray model. For the implementation of the model, we combinea two-step algorithm called Evolve-Filter (EF) with a computationally efficient finitevolume method. The main novelty of the proposed approach relies in applying spatialfiltering both for the collection of the snapshots and in the reduced order model, as wellas in considering the pressure field at reduced level. In both steps of the EF algorithm,velocity and pressure fields are approximated by using different POD basis and coeffi-cients. For the reconstruction of the pressures fields, we use a pressure Poisson equationapproach. We test our ROM on two benchmark problems: 2D and 3D unsteady flowpast a cylinder at Reynolds number 0 ≤ Re ≤ Reduced order models (ROMs) have been proposed as an efficient tool for the approximationof systems of parametrized partial differential equations, as they significantly reducing thecomputational cost required by classical full order models (FOMs), e.g. finite element methodsor finite volume methods. The basic ROM framework consists of two steps. First, a databaseof several solutions is collected by solving the original full order model for different physicaland/or geometrical configurations ( offline phase ). Then, the information obtained duringthe offline phase is used to compute the solution for newly specified values of the parametersin a short amount of time ( online phase ). For a comprehensive review on ROMs, we refer to[29, 46, 8, 7, 4, 9].It is well known that the extension to turbulent flows present several challenges. One ofthe reasons is that projection based ROMs of turbulent flows are affected by energy stabilityproblems [16]. This is related to the fact that proper orthogonal decomposition (POD)retains the modes biased toward large, high-energy scales, while the turbulent kinetic energy ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ m a t h . NA ] S e p s dissipated at level of the small turbulent scales [43]. A possible strategy to stabilize ROMsfor turbulent flows is the introduction of dissipation via a closure model [61, 3]. In [19], itwas shown theoretically and numerically that POD modes have a similar energy transfermechanism to Fourier modes. Therefore, the use of Large Eddy Simulation (LES) couldbe beneficial. Data-driven ROMs for LES full order models have been successfully used forhydroacoustic analysis [23]. In addition, the efficiency of ROMs for LES-VMS stabilizedfinite elements has been proved through relevant numerical benchmarks [52].We focus on projection based ROMs (see, e.g., [2]) that have been successfully applied toseveral fluid dynamics problems. We propose a POD-Galerkin-based ROM for a LES filteringapproach. We consider a variant of the so-called Leray model [40], where the small-scaleeffects are described by a set of equations to be added to the discrete Navier-Stokes equations.This extra problem acts as a differential low-pass filter [13]. For its actual implementation, weuse the Evolve-Filter (EF) algorithm [14, 22, 21, 39]. The Leray model has been extensivelyapplied within a Finite Element framework, while we focus on Finite Volume (FV) methods[26]. In the ROM context, the Leray model has been applied to the 1D Kuramoto-Sivashinskyequations [51], stochastic Burgers equation [31], and Navier-Stokes equations [28, 66, 63]. TheEF algorithm has also been investigated in combination with regularized ROMs: applicationsinclude stochastic Burgers equation [64] and 3D Navier-Stokes equations [63]. In [27], arelaxation step is added to the EF algorithm and applied to the 2D Navier-Stokes equations,while in [65] a ROM is applied to the approximate deconvolution model. We note that in [28,66, 63, 27, 65] LES filtering is used in the development of the ROM to address the inaccuracyand numerical instability of the standard Galerkin ROM for convection-dominated problems.Unlike those works, we use LES filtering also as FOM, i.e. to generate the snapshot data.Such an approach ensures a greater consistency between FOM and ROM. The strategy weproposed is mentioned in [65] as a future perspective. However, to the best of our knowledgeit has not been attempted so far.The novelties of our approach include:- the computation of the pressure field at ROM level (through the pressure Poissonequation [53, 55, 57]);- the use of different POD coefficients and bases to approximate the two velocity fieldsin the Leray model.We test our approach on two benchmarks: 2D [59, 33] and 3D [59] flow past a cylinderwith time-dependent Reynolds number 0 ≤ Re ( t ) ≤ The full order model
We consider a fixed domain Ω ⊂ R D with D = 2 , t , T ) ⊂ R + . Let π ∈ P ⊂ R d be a parameter vector in a d -dimensional parameter space P . Theso-called Leray model couples the Navier-Stokes equations (NSE) with a differential filter: ρ ∂ t u ( x , t ; π ) + ρ ∇ · ( u ( x , t ; π ) ⊗ u ( x , t ; π )) − µ ∆ u ( x , t ; π ) + ∇ p ( x , t ; π ) = 0 , (1) ∇ · u ( x , t ; π ) = 0 , (2) − α ∆ u ( x , t ; π ) + u ( x , t ; π ) + ∇ λ ( x , t ; π ) = u ( x , t ; π ) , (3) ∇ · u ( x , t ; π ) = 0 , (4)in Ω × ( t , T ), endowed with the boundary conditions. In (1)-(4), ρ is the fluid density, µ isthe dynamic viscosity, u is velocity, p is the pressure, u is the filtered velocity , and variable λ is a Lagrange multiplier to enforce the incompressibility constraint for u . Parameter α canbe interpreted as the filtering radius . Problem (1)-(4) is endowed with suitable boundary u ( x , t ; π ) = u D ( x , t ; π ) on ∂ Ω D × ( t , T ) , (5)(2 µ ∇ u ( x , t ; π ) − p ( x , t ; π ) I ) n = on ∂ Ω N × ( t , T ) , (6) u ( x , t ; π ) = u D ( x , t ; π ) on ∂ Ω D × ( t , T ) , (7)(2 α ∇ u ( x , t ; π ) − λ ( x , t ; π ) I ) n = on ∂ Ω N × ( t , T ) . (8)and the initial data u ( x , t ) = u ( x ) in Ω × { t } . Here ∂ Ω D ∪ ∂ Ω N = ∂ Ω and ∂ Ω D ∩ ∂ Ω N = ∅ .In addition, u D and u are given.For the sake of simplicity, from now on the dependance of the variables on x , t , andparameter π will be omitted. We start with the time discretization of the Leray model (1)-(4). Let ∆ t ∈ R , t n = t + n ∆ t ,with n = 0 , ..., N T and T = t + N T ∆ t . Moreover, we denote by y n the approximation of ageneric quantity y at the time t n . We adopt a Backward Differentiation Formula of order 2(BDF2) [47].To decouple the Navier-Stokes system (1)-(2) from the filter system (3)-(4) at each timestep, we consider the Evolve-Filter (EF) algorithm [14, 22, 21]. This algorithm reads asfollows: given the velocities u n − and u n , at t n +1 :i) evolve : find intermediate velocity and pressure ( v n +1 , q n +1 ) such that ρ t v n +1 + ρ ∇ · (cid:0) u ∗ ⊗ v n +1 (cid:1) − µ ∆ v n +1 + ∇ q n +1 = b n +1 , (9) ∇ · v n +1 = 0 , (10)with boundary conditions v n +1 = u n +1 D on ∂ Ω D × ( t , T ) , (11)(2 µ ∇ v n +1 − q n +1 I ) n = on ∂ Ω N × ( t , T ) , (12)and initial condition v = u in Ω × { t } . In eq. (9), we set u ∗ = 2 u n − u n − and b n +1 = (4 u n − u n − ) / (2∆ t ). 3i) filter : find ( u n +1 , λ n +1 ) such that − α ∆ u n +1 + u n +1 + ∇ λ n +1 = v n +1 , (13) ∇ · u n +1 = 0 , (14)with boundary conditions u n +1 = u n +1 D on ∂ Ω D × ( t , T ) , (15)(2 α ∇ u n +1 − λ n +1 I ) n = on ∂ Ω N × ( t , T ) . (16)We consider u n +1 and q n +1 the approximation of the velocity and the pressure at the time t n +1 , respectively.In this work, we consider only homogeneous Neumann boundary conditions. For thetreatment of non-homogeneous boundary conditions, we refer to [11]. Remark 2.1.
Filter problem (13) - (14) can be considered a generalized Stokes problem. Infact, if we divide eq. (13) by ∆ t and rearrange the terms we obtain: ρ ∆ t u n +1 − µ ∆ u n +1 + ∇ q n +1 = ρ ∆ t v n +1 , µ = ρ α ∆ t , (17) where the filtered pressure q n +1 = ρλ n +1 / ∆ t . Problem (17) , (14) can be seen as a timedependent Stokes problem with viscosity µ , discretized by the Backward Euler (or BDF1)scheme. A solver for problem (17) , (14) can then be obtained by adapting a standard linearizedNavier-Stokes solver. For the space discretization of problems (9)-(10) and (17),(14), we adopt a Finite Volume(FV) method. We partition the computational domain Ω into cells or control volumes Ω i ,with i = 1 , . . . , N c , where N c is the total number of cells in the mesh. Let A j be the surfacevector of each face of the control volume, with j = 1 , . . . , M .The fully discretized form of problem (9)-(10) is given by ρ t v n +1 i + ρ (cid:88) j ϕ ∗ j v n +1 i,j − µ (cid:88) j ( ∇ v n +1 i ) j · A j + (cid:88) j q n +1 i,j A j = b n +1 i , (18) (cid:88) j ( ∇ q n +1 ) j · A j = (cid:88) j ( H ( v n +1 i )) j · A j , (19)where: H ( v n +1 i ) = − ρ (cid:88) j ϕ ∗ j v n +1 i,j + 2 µ (cid:88) j ( ∇ v n +1 i ) j · A j + b n +1 i with ϕ ∗ j = u ∗ j · A j . (20)In (18)-(20), v n +1 i and b n +1 i denotes the average velocity and source term in control volumeΩ i , respectively. Moreover, we denote with v n +1 i,j and q n +1 i,j the velocity and pressure associatedto the centroid of face j normalized by the volume of Ω i .The fully discrete problem associated to the filter (17),(14) is given by ρ ∆ t u n +1 i − (cid:88) j µ j ( ∇ u n +1 i ) j · A j + (cid:88) j q n +1 i,j A j = ρ ∆ t v n +1 i , (21) (cid:88) j ( ∇ q n +1 i ) j · A j = (cid:88) j ( H ( u n +1 i )) j · A j , (22)4ith H ( u n +1 i ) = (cid:88) j µ j ( ∇ u n +1 i ) j · A j + ρ ∆ t v n +1 . (23)In (21)-(23), we denoted with u n +1 i the average filtered velocity in control volume Ω i , while q n +1 i,j is the auxiliary pressure at the centroid of face j normalized by the volume of Ω i . Formore details, we refer the reader to [26].We have implemented the EF algorithm within the C++ finite volume library OpenFOAM R (cid:13) [62]. For the solution of the linear system associated with (18)-(19) we used the PISO algo-rithm [32], while for problem (21)-(22) we chose a slightly modified version of the SIMPLEalgorithm [45], called SIMPLEC algorithm [60]. Both PISO and SIMPLEC are partitionedalgorithms that decouple the computation of the pressure from the computation of the ve-locity.The approach described in this section represents our Full Order Model (FOM). The Reduced Order Model (ROM) we propose is an extension of the model introduced in[53, 55]. In Sec 3.1 we introduce the procedure we use to construct a POD-Galerkin ROM andin Sec. 3.2 we present the strategy we choose for pressure stabilization at reduced order level.Finally, Sec. 3.3 describes the lifting function method we apply to enforce non-homogeneousDirichlet boundary conditions for the velocity field at the reduced order level. The ROMcomputations are carried out using ITHACA-FV [54], an in-house open source C++ library.
The main idea of reduced order modeling for parametrized PDEs is the assumption thatsolutions live in a low dimensional manifold. Thus, any solution can be approximated as alinear combination of a reduced number of global basis functions.We approximate velocity fields v and u and pressure fields q and q as linear combinationsof the dominant modes (basis functions), which are assumed to be dependent on space vari-ables only, multiplied by scalar coefficients that depend on the time and/or the parameters: v ≈ v r = N rv (cid:88) i =1 β i ( π , t ) ϕ i ( x ) , q ≈ q r = N rq (cid:88) i =1 γ i ( π , t ) ψ i ( x ) , (24) u ≈ u r = N ru (cid:88) i =1 β i ( π , t ) ϕ i ( x ) , q ≈ q r = N rq (cid:88) i =1 γ i ( π , t ) ψ i ( x ) . (25)In (24)-(25), N r Φ denotes the cardinality of a reduced basis for the space field Φ belongs to.We point out that, unlike previous works [28, 63, 66, 27], we compute also pressurefields q and q . Furthermore, we consider different basis and different coefficients for theapproximation of the velocity fields v and u . This follows from the fact that we apply thefiltering step for both FOM and ROM.In the literature, one can find several techniques to generate the reduced basis spaces,e.g. Proper Orthogonal Decomposition (POD), the Proper Generalized Decomposition (PGD)and the Reduced Basis (RB) with a greedy sampling strategy. See, e.g., [49, 17, 36, 46, 18,20, 58, 9]. We find the reduced basis by using the method of snapshots.5et K = { π , . . . , π N k } be a finite dimensional training set of samples chosen inside theparameter space P and for each time instance t k ∈ { t , . . . , t N t } ⊂ ( t , T ]. We solve the FOMdescribed in Sec. 2 for each π k ∈ K ⊂ P . The total number of snapshots N s is given by N s = N k · N t . The snapshots matrices are obtained from the full-order snapshots: S Φ = [Φ( π , t ) , . . . , Φ( π N k , t N t )] ∈ R N h Φ × N s for Φ = { v h , u h , q h , q h } , (26)where the subscript h denotes a solution computed with the FOM and N h Φ is the dimensionof the space Φ belong to in the FOM. Note that Φ could be either a scalar or a vectorfield. The POD problem consists in finding, for each value of the dimension of the PODspace N P OD = 1 , . . . , N s , the scalar coefficients a , . . . , a N s , . . . , a N s , . . . , a N s N s and functions ζ , . . . , ζ N s , that minimize the error between the snapshots and their projection onto the PODbasis. In the L -norm, we have E N POD = arg min N s (cid:88) i =1 || Φ i − N POD (cid:88) k =1 a ki ζ k || ∀ N P OD = 1 , . . . , N s with ( ζ i , ζ j ) L (Ω) = δ i,j ∀ i, j = 1 , . . . , N s . (27)It can be shown [37] that eq. (27) is equivalent to the following eigenvalue problem C Φ Q Φ = Q Φ Λ Φ , (28) C Φ ij = (Φ i , Φ j ) L (Ω) for i, j = 1 , . . . , N s , (29)where C Φ is the correlation matrix computed from the snapshot matrix S Φ , Q Φ is the matrixof eigenvectors and Λ Φ is a diagonal matrix whose diagonal entries are the eigenvalues of C Φ .Then, the basis functions are obtained as follows: ζ i = 1 N s Λ Φ i N s (cid:88) j =1 Φ j Q Φ ij . (30)The POD modes resulting from the aforementioned methodology are: L Φ = [ ζ , . . . , ζ N r Φ ] ∈ R N h Φ × N r Φ , (31)where N r Φ < N s are chosen according to the eigenvalue decay of the vectors of eigenvalues Λ . The reduced order model can be obtained through a Galerkin projection of the governingequations onto the POD spaces.Let M r ij = ( ϕ i , ϕ j ) L (Ω) , (cid:102) M r ij = ( ϕ i , ϕ j ) L (Ω) , A r ij = ( ϕ i , ∆ ϕ j ) L (Ω) , (32) B r ij = ( ϕ i , ∇ ψ j ) L (Ω) , P r ij = ( ψ i , ∇ · ϕ j ) L (Ω) , (33)where ϕ i and ψ i are the basis functions in (24). The reduced algebraic system at time t n +1 for problem (9)-(10) is: ρ t M r β n +1 + ρ G r ( β n , β n − ) β n +1 − µ A r β n +1 + B r γ n +1 = ρ ∆ t (cid:102) M r (cid:18) β n − β n − (cid:19) , (34) P r β n +1 = 0 , (35)6here vectors β n +1 and γ n +1 contain the values of coefficients β i and γ i in (24) at time t n +1 .The term G r ( β n , β n − ) β n +1 in (34) is related to the non-linear convective term: (cid:16) G r ( β n , β n − ) β n +1 (cid:17) i = (2 β n − β n − ) T G r i ..β n +1 (36)where G r is a third-order tensor defined as follows [47, 48] G r ijk = ( ϕ i , ∇ · ( ϕ j ⊗ ϕ k )) L (Ω) . (37)Next, let M r ij = ( ϕ i , ϕ j ) L (Ω) , A r ij = ( ϕ i , ∆ ϕ j ) L (Ω) , (38) B r ij = ( ϕ i , ∇ ψ j ) L (Ω) , P r ij = ( ψ i , ∇ · ϕ j ) L (Ω) , (39)where ϕ i and ψ i are the basis functions in (25). The reduced algebraic system at time t n +1 for problem (17),(14) is ρ ∆ t M r β n +1 − µ A r β n +1 + B r γ n +1 = ρ ∆ t (cid:102) M Tr β n +1 , (40) P r β n +1 = 0 . (41)where vectors β n +1 and γ n +1 contain the values of coefficients β i and γ i in (25) at time t n +1 .The complete reduced algebraic system at time t n +1 is given by (34)-(35),(40)-(41). Fi-nally, the initial conditions for the ROM algebraic system (34)-(35) are obtained performinga Galerkin projection of the initial full order condition onto the POD basis spaces: β i = ( v ( x , π , t ) , ϕ i ) L (Ω) ,β i = ( u ( x , π , t ) , ϕ i ) L (Ω) . For the accurate reconstruction of the pressure field at the reduced level, different approacheshave been proposed. One option is to use a global POD basis for both pressure and ve-locity field and same temporal coefficients [10, 42]. Another option to satisfy the inf-sup(or Ladyzhenskaya-Brezzi-Babuska) condition [15, 12] is through the supremizer enrichmentmethod [50, 5, 55]. Finally, one can take the divergence of the momentum equation to obtaina Poisson equation for pressure that is projected onto a POD basis [1]. This method, calledPoisson pressure equation (PPE), has been recently extended to a finite volume context[53, 55, 56, 57].We choose to adopt and extend the PPE method used in [53, 55, 57] for both pressurefields in the EF algorithm. We take the divergence of eq. (9) and (17) and account fordivergence free conditions (10) and (14) to obtain the Poisson pressure equation:∆ q n +1 = − ρ ∇ · (cid:0) ∇ · (cid:0) u ∗ ⊗ v n +1 (cid:1)(cid:1) , (42)∆ q n +1 = 0 , (43)with boundary conditions (11), (15), and: ∂ n q n +1 = − µ n · (cid:0) ∇ × ∇ × v n +1 (cid:1) − n · (cid:18) ρ t v n +1 − b n +1 (cid:19) on ∂ Ω N × ( t , T ) , (44) ∂ n q n +1 = − µ n · (cid:0) ∇ × ∇ × u n +1 (cid:1) on ∂ Ω N × ( t , T ) , (45)7here ∂ n denotes the derivative with respect to the normal vector n . So, at reduced level,instead of solving (9)-(10) and (17),(14), we solve modified systems (9), (42) and (17), (43).For the enforcement of non-homogeneous Neumann conditions for the pressure fields, werefer to [44, 35]. Remark 3.1.
Systems (9) - (10) and (17) , (14) are not equivalent to systems (9) , (42) and (17) , (43) for steady flows [41, 44, 35]. However, as discussed in Remark 2.1, the filterproblem can be considered as a time-dependent Stokes problem. The matrix form of eq. (42) and (43) is obtained, as usual, after integrating by parts theLaplacian terms in the weak formulation and accounting for the boundary conditions. Weobtain: D r γ n +1 + ρ J r ( β n , β n − ) β n +1 − µ N r β n +1 − ρ t F r = 0 , (46) D r γ n +1 − µ N r β n +1 = 0 , (47)where D r ij = ( ∇ ψ i , ∇ ψ j ) L (Ω) , (48) N r ij = ( n × ∇ ψ i , ∇ × φ j ) L ( ∂ Ω) , F r ij = ( ψ i , n · (3 ϕ j − ϕ j + ϕ j )) L ( ∂ Ω) , (49) D r ij = ( ∇ ψ i , ∇ ψ j ) L (Ω) , N r ij = ( n × ∇ ψ i , ∇ × φ j ) L ( ∂ Ω) . (50)The residual associated with the non-linear term in the equation (46) is evaluated usingthe same strategy proposed for eq. (34). We have (cid:16) J r ( β n , β n − ) β n +1 (cid:17) i = (2 β n − β n − ) T J r i ..β n +1 , (51)where J r is a third-order tensor defined as follows J r ijk = ( ∇ ψ i , ∇ · ( ϕ j ⊗ ϕ k )) L (Ω) . (52)Finally, the ROM algebraic system that has to be solved at every time step is (34), (46),(40), (47). In order to homogeneize the velocity fields snapshots and make them independent on theboundary conditions, we use the lifting function method [53]. The lifting functions, alsocalled control functions, are problem-dependent: they have to be divergence free in orderto retain the divergence-free property of the basis functions and they have to satisfy theboundary conditions of the FOM. The velocity snapshots are then modified according to: v (cid:48) h = v h − N BC (cid:88) j =1 v BC j ( π , t ) χ j ( x ) , (53) u (cid:48) h = u h − N BC (cid:88) j =1 u BC j ( π , t ) χ j ( x ) , (54)where N BC is the number of non-homogeneous Dirichlet boundary conditions, χ ( x ) are thecontrol functions, and v BC j and u BC j are suitable temporal coefficients. The POD is applied8o the snapshots satisfying the homogeneous boundary conditions and then the boundaryvalue is added back: v r = N BC (cid:88) j =1 v BC j ( π , t ) χ j ( x ) + N rv (cid:88) i =1 β i ( π , t ) ϕ i ( x ) , (55) u r = N BC (cid:88) j =1 u BC j ( π , t ) χ j ( x ) + N ru (cid:88) i =1 β i ( π , t ) ϕ i ( x ) . (56) We consider two well-known test cases [33, 59]: 2D and 3D flow past a cylinder. We inves-tigate the performance of the ROM model in the reconstruction of the time evolution of theflow field. For the 2D case, parametrization of the filtering radius is also introduced.
This benchmark has been thoroughly investigated at FOM level in a finite volume envi-ronment in [26]. To the best of our knowledge, it is the first time that this benchmark isconsidered within a ROM framework.The computational domain is a 2.2 × ρ = 1 and viscosity µ = 10 − . We impose a no slip boundary conditionon the upper and lower wall and on the cylinder. At the inflow, we prescribe the followingvelocity profile: v (0 , y, t ) = (cid:18) . sin ( πt/ y (0 . − y ) , (cid:19) , y ∈ [0 , . , t ∈ (0 , , (57)and ∂q/∂ n = ∂q/∂ n = 0. At the outflow we prescribe ∇ v · n = 0 and q = q = 0. We startthe simulations from fluid at rest. Note that the Reynolds number is time dependent, with0 ≤ Re ≤
100 [59].Figure 1: 2D flow past a cylinder: (left) part of the mesh under consideration and (right)the lifting function for velocity.We consider a hexaedral computational grid with h min = 4 . e − h max = 1 . e − . e ◦ ), average non-orthogonality (4 ◦ ), skewnwss (0.7), andmaximum aspect ratio (2). Fig. 1 (left) shows a part of the mesh. We chose this mesh becauseit is the coarsest among all the meshes considered in [26] and thus the most challenging forour filtering approach.While in [26] the choice of the time step depended on the Courant-Friedrichs-Lewy number( CF L max ) set to
CF L max = 0 .
2, in this work we consider a fixed time step for sake of9onsistency with the ROM. In order to obtain comparable solutions, we set ∆ t = 4 e − CF L max = 0 . χ (0 , y ) = (cid:18) . y (0 . − y ) , (cid:19) , y ∈ [0 , . , (58)and uniform null values the rest of the boundary. See Figure 1 (right).We will compare our findings mainly with [53, 55, 57], because these references developeda NSE-ROM finite volume framework with a PPE stabilization method for the reconstructionof the pressure field. Moreover, the benchmarks presented in [53, 55, 57] share some featureswith the ones we consider: a 2D flow past cylinder at Re = O (100) [53, 55] (although with asteady and uniform inflow condition), and time-dependent uniform inflow boundary condition[57] (although in a Y-junction flow problem).We are going to investigate the accuracy of our reduced order model with respect to thetime history of the velocity and pressure fields. We set α = 0 . C Φ in (28) is 800 ×
800 and N us = N vs = N qs = N qs = 800. Following [24], we performed a convergence test as the numberof snapshots increases. We multiplied the frequency at which the snapshots are collected by 5and 10, leading to N us = N vs = N qs = N qs = 160 and N us = N vs = N qs = N qs = 80, respectively.We calculated the L relative error: E Φ = || Φ h ( t, π ) − Φ r ( t, π ) || L (Ω) || Φ h ( t, π ) || L (Ω) , (59)where Φ h and Φ r are a particular field computed with the FOM (i.e., v h , u h , q h or q h ) andthe ROM (i.e., v r , u r , q r or q r ), respectively. Moreover, we evaluate the relative error of thekinetic energy E K Φ h = | K Φ h ( t, π ) − K Φ r ( t, π ) | K Φ h ( t, π ) , (60)where K Φ h and K Φ r are the values of the kinetic energy computed by the FOM (for v h or u h ) and by the ROM (for v r or u r ), respectively.Fig. 2 shows error (59) for all the velocity and pressure fields and error (60) over timefor the three different sampling frequencies. Fig. 2 shows that, except for the initial and thefinal time of the simulation, the relative error for velocities fields is significantly lower than10 − over most of the time interval and reaches a minimum value of 7 . · − . Moreover,we notice that there is no substantial difference in the errors for the different samplingfrequencies. Thus, to reduce the computational cost of the offline phase, we will consider thelargest sampling frequency (i.e., 0.1) for the results presented from here on.We report minimum, maximum and average (over time) relative errors in Table 1. Weremark that the average errors for the velocity fields are comparable to the values (between10 v q q K u K v Maximum E Φ E Φ E Φ u v q q . · − and 3 · − ) obtained for steady flow past cylinder Re = 100 [55]. Larger velocityerrors during the first few time steps might be due to the transient nature of the flow, as alsonoted for lid driven cavity flow studied in [57]. As for pressure q , the relative error reachesits maximum value far from the end points of the time interval, while the average error for q is comparable to the errors in [55] and [57]. The error for q stays close to the average duringmost of the time interval. Finally, we observe that the smallest relative errors are achievedfor the kinetic energies of the system.Table 2 lists the first 3 cumulative eigenvalues, based on the first 15 most energetic PODmodes. We see that 2 modes for u , v and q , and 1 mode for q, are sufficient to retain 99.99% ofthe energy contained in the snapshots. It has been verified that adding more modes does notincrease the accuracy of the ROM results. These first (homogenized) velocity and pressuremodes are plotted in Figure 3. Remark 4.1.
Fig. 4 displays the difference between the first two POD modes for velocities u and v . We note that there are significant differences right next to the cylinder where thefiltering stabilization plays a key role. This justifies the fact to consider different bases toapproximate the two velocity fields. We report in Fig. 5 the homogeneized velocity u computed at times t = 0 . , , . ,
8. Weobserve that up to t = 7 . t = 8 it isnot comparable with either the first or the second POD mode. This could explain the largerrelative errors for u towards the end of the time interval shown in Fig. 2.Figs. 6 and 7 display the comparison between the computed FOM and ROM fields at twodifferent times, t = 1 and t = 5. As one can see from the figures, the ROM provides a goodglobal reconstruction of both velocities and both pressures. Next, we make the comparisonmore quantitative. Fig. 8 displays the difference between the computed FOM and ROM fieldsfor t = 1 and t = 5. The maximum absolute error between the FOM and the ROM for thevelocity fields is of the order of 10 − at t = 1, while it reaches lower values (of order 10 − )at t = 5. The maximum absolute error for q is of the order of 10 − at t = 1 ,
5. Again, thesevalues are in perfect agreement with those reported for the lid driven cavity and Y-junction11igure 2: 2D flow past a cylinder: time history of L norm of the relative error for thevelocity fields (top), pressure fields (center), and for kinetic energies of the system (bottom)for different numbers of snapshots. We consider 2 modes for v , u and q , and 1 mode for ¯ q .12 irst mode for v Second mode for v First mode for u Second mode for u First mode for q Second mode for q First mode for q Figure 3: 2D flow past a cylinder: first 2 POD modes for velocity v (1nd raw), velocity u (2nd raw), pressure q (3nd raw) and first POD mode for pressure q (4nd raw).Figure 4: 2D flow past a cylinder: difference between the first (left) and the second (right)POD modes related to the velocities u and v . u F OM u F OM u F OM u F OM
Figure 5: 2D flow past a cylinder: FOM (homogeneized) velocity field u at t = 0 .
1, 5 (firstraw, from left to right), 7 . F OM u ROM v F OM v ROM q F OM q ROM q F OM q ROM
Figure 6: 2D flow past a cylinder: comparison between FOM and ROM u (first row), v (second row), q (third row), and q (fourth row) at time t = 1. We consider 2 modes for v , u and q , and 1 mode for ¯ q .flow in [57] (no absolute error is reported in [53, 55]) and they indicate that our ROM is ableto reproduce the main flow features at different times. Finally, we note that the maximumabsolute error for q is of the order of 10 − at t = 1 and 1 at t = 5. However, we note that at t = 5 the order of magnitude of q is 10, as shown in Fig. 7.To further quantitate the accuracy of the ROM with respect to the FOM, we considerthe quantities of interest for this benchmark, i.e. the drag and lift coefficients [33, 59]: c d ( t ) = 2 ρL r U r (cid:90) S ((2 µ ∇ u − q I ) · n ) · t dS, c l ( t ) = 2 ρL r U r (cid:90) S ((2 µ ∇ u − q I ) · n ) · n dS, (61)where U r = 1 is the maximum velocity at the inlet/outlet, L r = 0 . S is the cylinder surface, and t and n are the tangential and normal unit vectors to the cylinder,respectively. The FOM/ROM comparison for the coefficients in (61) over time is reported inFig. 9. We observe that the amplitude of the force coefficients are slightly underestimated forall the time instants by ROM. The ROM reconstruction of the lift coefficient appears to bemore critical, especially around the center of the time interval. This could be due to the factthat larger errors for pressure q are localized close to the cylinder, as one can see in Fig. 8(third row). 14 F OM u ROM v F OM v ROM q F OM q ROM q F OM q ROM
Figure 7: 2D flow past a cylinder: comparison between FOM and ROM u (first row), v (second row), q (third row), and q (fourth row) at time t = 5. We consider 2 modes for v , u and q , and 1 mode for ¯ q . 15 F OM − v ROM , t = 1 v F OM − v ROM , t = 5 u F OM − u ROM , t = 1 u F OM − u ROM , t = 5 q F OM − q ROM , t = 1 q F OM − q ROM , t = 5 q F OM − q ROM , t = 1 q F OM − q ROM , t = 5Figure 8: 2D flow past a cylinder: difference between FOM and ROM v (first row), u (secondrow), q (third row), and q (fourth row) at times t = 1 (left) and t = 5 (right). We consider2 modes for v , u and q , and 1 mode for ¯ q .Figure 9: 2D flow past a cylinder: aerodynamic coefficients C d (left) and C l (right) computedby FOM and ROM. We consider 2 modes for v , u and q , and 1 mode for ¯ q .16 ( c l,max ) c l,max t ( c d,max ) c d,max FOM 4.1 0.258 3.9 1.135ROM 4.1 0.235 3.9 1.041Table 3: 2D flow past a cylinder: maximum lift and drag coefficients computed by FOM andROM and times at which the maxima occur. We recall that the sampling frequency is 0.1.Table 3 compares the maximum lift and drag coefficients and times at which the maximaoccur. In addition, we obtain the following errors: E c d = c F OMd,max − c ROMd,max c F OMd,max = 0 . , E t cd = t F OMc d,max − t ROMc d,max t F OMc d,max = 0 , (62) E c l = c F OMl,max − c ROMl,max c F OMl,max = 0 . , E t cl = t F OMc l,max − t ROMc l,max t F OMc l,max = 0 . (63)We see that our ROM approach is able to provide a perfect prediction of the time istants wheremaxima values of the aerodynamic coefficients occur. The errors related to the maximumvalues are both lower than 9%. In [53], the authors use the NSE model with no filteringfor the steady flow at Re = 100 around a cylinder and find errors smaller than 5% for bothcoefficients. Additional differences, such as steady boundary conditions and homogeneousNeumann boundary condition for pressure in [53], make it harder to identify the reason whywe obtain larger errors.Regarding the computational cost, the CPU time of the FOM model is 3800 s. The CPUtime of the ROM is 30 s. This corresponds to a speed-up of ≈ α Filtering radius α plays a crucial role in the success of filtering stabilization. So, after havinginvestigated the ability of our ROM approach to reconstruct the time evolution of velocityand pressure fields, we consider α as a parameter. To train the ROM, we choose a uniformsample distribution in the range α ∈ [0 . , h min ], where 0 . Re − / L r . We firstconsider 11 sampling points and then decrease to 6 sampling points. For each value of thefiltering radius inside the training set, a simulation is run for the entire time interval ofinterest, i.e. (0 , α = 0 . α = 0 . α = 0 . α = 0 . α = 0 . t = 1 and t = 5. Again, we see that our ROM approach is able to reproduce the main17igure 10: 2D flow past a cylinder: time history of L norm of the relative error for velocityfields (top), pressure fields (center), and for kinetic energies of the system (bottom) for adifferent number of collected snapshots. We consider 2 modes for v , u and q , and 1 modefor ¯ q . u v q q K u K v Maximum E Φ E Φ E Φ α = 0 . F OM − v ROM , t = 1 v F OM − v ROM , t = 5 u F OM − u ROM , t = 1 u F OM − u ROM , t = 5 q F OM − q ROM , t = 1 q F OM − q ROM , t = 5 q F OM − q ROM , t = 1 q F OM − q ROM , t = 5Figure 11: 2D flow past a cylinder: Difference between FOM and ROM v (first row), u (second row), q (third row), and q (fourth row) at times t = 1 (left) and t = 5 (right) for α = 0 . v , u and q , and 1 mode for ¯ q .flow features at different times, with a level of accuracy that is comparable to the case wheretime is the only parameter.The FOM/ROM comparison for the drag and lift coefficients (61) over time for α =0 . α the coefficients com-puted by the ROM underestimate the coefficients computed by the FOM over almost theentire time interval under consideration, while the phase is perfectly reproduced. Table 5reports quantitative comparison between ROM and FOM in terms of maximum lift and dragcoefficients and times at which the maxima occur. The corresponding relative error, as de-fined in (62)-(63), are: E c l = 0 . E t cl = 0, E c d = 0 . E t cd = 0. In switching from α = 0 . α = 0 . C d error decreases for (from 8.3% to 6%), while the C d error increases (from 8.9% to 12.6%). The increase in the errore for C l could be due to thefact that towards the center of the time interval the absolute error for pressure q around thecylinder is slightly larger for α = 0 . t ( c l,max ) c l,max t ( c d,max ) c d,max FOM 4.1 0.294 3.9 1.025ROM 4.1 0.257 3.9 0.963Table 5: 2D flow past a cylinder: maximum lift and drag coefficients, and times at which themaxima occur for FOM and ROM, for α = 0 . C d (left)and C l (right), and the corresponding ROM reconstructions, for α = 0 . v , u and q , and 1 mode for ¯ q . In this section, we aim at showing that our ROM approach can easily handle three-dimensionalproblems. The 3D benchmark we consider has been studied for the first time in [59] and fur-ther investigated in [6, 34].We choose to adopt the classical Leray model, instead of the EF algorithm, which isknow to be over-diffusive especially in 3D. As a regularized ROM for convection-dominatedproblems, the Leray model has been investigated in [51, 31, 28, 66, 63]. The discrete in timemodel reads as follows: Given velocities u n − and u n , at t n +1 find u n +1 , q n +1 , u n +1 and q n +1 such that: ρ u n +1 − u n + u n − t + ρ ∇ · (cid:0) u ∗ ⊗ u n +1 (cid:1) − µ ∆ u n +1 + ∇ q n +1 = 0 , (64) ∇ · u n +1 = 0 , (65) ρ ∆ t u n +1 − µ ∆ u n +1 + ∇ q n +1 = ρ ∆ t u n +1 , (66) ∇ · u n +1 = 0 , (67)where u ∗ = 2 u n − u n − . Also with this model, we compute pressure fields and apply thefilter for both FOM and ROM. However, for sake of brevity we will show results for u and q only.The computational domain is a 2.5 × × z -axis and center is located at (0.5, 0.2) when taking the bottom leftcorner of the channel as the origin of the axes. Fig. 13 (left) shows part of the computationaldomain. The channel is filled with fluid with density ρ = 1 and viscosity µ = 0 . u (0 , y, z, t ) = (cid:18) . sin ( πt/ yz (0 . − y ) (0 . − z ) , , (cid:19) , y, z ∈ [0 , . , t ∈ (0 , . (68)In addition, on the channel walls, cylinder, and at the inlet we impose ∂q/∂ n = ∂q/∂ n = 0where n is the outward normal. At the outflow, we prescribe ∇ u · n = 0 and q = q = 0.Note that the Reynolds number is time dependent, with 0 ≤ Re ≤
100 [59, 6, 34]. Like forthe 2D benchmark, we start the simulations from fluid at rest.20igure 13: 3D flow past a cylinder: (left) part of the mesh under consideration and (right)evolution of c l computed with the Leray model and NSE model. C d C l NSE algorithm 3.21 0.004Leray model 3.12 0.0028Ref. values [59, 6, 34] [3.2, 3.3] [0.002, 0.004]Table 6: 3D flow past a cylinder: Maximum drag and lift coefficients given by NSE and Leraymodel. The bottom row reports the reference values from [59, 6, 34].We consider a hexahedral grid with h min = 9 e − h max = 6 . e − ◦ ), average non-orthogonality (7 ◦ ), skewnwss(0.6), and maximum aspect ratio (25). In addition, the mesh is refined next to the cylinder,like the meshes used in [6, 34].Before applying the ROM, we test the Leray model and compare its results with theresults produced by a NSE solver. Like in the 2D case, we use a second-order accurateCentral Differencing (CD) scheme [38] for the discretization of the convective term. For theNSE solver, we set ∆ t = 1 e −
4, while for the Leray model we choose ∆ t = 5 e −
3. The largertime step for the Leray model is motivated by an attempt to contain the artificial viscosity¯ η defined in eq. (17). We set α = 0 . c l over time computed by the Leray and NSE model. Weobserve that the Leray model dampens the unphysical oscillations in the NSE solution andreduces the maximum lift coefficient. We report in Table 6 the computed values of maximumdrag and lift coefficients, together with the reference values fom [59, 6, 34]. From Fig. 13(right), we can conclude that the Leray model outperforms the NSE model on a coarse mesh.The snapshots are collected every 0.1 s using an equispaced grid method in time. Fig. 14shows error (59) for the velocity u and pressure q field over time. As for the 2D case, thelargest relative errors for u occur around the beginning and end of the simulation, whilethe relative error for pressure q reaches its maximum value around t = 5. The minimum,maximum and average (over time) relative errors are reported in Table 7. Average errorsfor both the velocity and pressure field are comparable with the ones found for the 2Dbenchmark. 21igure 14: 3D flow past a cylinder: time history of L -norm of the relative error for velocity u (left) and pressure q (right). We consider 9 modes for u and 4 modes for q . u q Maximum E Φ E Φ E Φ u and pressure q .Table 8 lists the first 10 cumulative eigenvalues, based on the first 15 most energeticPOD modes. We see that 9 modes for u and 4 modes for q are sufficient to retain 99.99%of the energy contained in the snapshots. With respect to the 2D case, a larger number ofmodes is necessary. This could be due to the fact that the EF algorithm used for the 2D testintroduces more numerical dissipation and dampens high frequency modes.Fig. 15 displays the difference between the solutions on the midsection ( z = 0 . t = 5. The maximum absolute errors between the FOMand the ROM for both u and q are of the order 0 .
1. In general, we can still conclude thatour ROM approach is able to reproduce the main flow features.We report in Fig. 16 the evolution of the lift and drag coefficients computed by FOMMode number u q u and q .22igure 15: 3D flow past a cylinder: difference between FOM and ROM u (left) and q (right)on the midsection at time t = 5. We consider 9 modes for u and 4 modes for q .and ROM with two different values of N rq . We see that the amplitude of the drag coefficientis reproduced with excellent accuracy over the entire time interval. However, the ROMreconstruction of the lift coefficient appears to be more critical, as already observed in the2D test. Again, this could be due to the fact that larger errors for pressure q are localizedclose to the cylinder, as one can see in Fig. 15 (right). Table 9 compares the maximum liftand drag coefficients given by ROM and FOM. The relative errors related for the maximumvalues are E c d = 0 .
3% and E c l = 17 . C l isobtained by using a higher number of modes for the pressure (i.e., larger N rq ) in order toreconstruct more accurately the low amplitude oscillatory pattern around the time the peakis reached. See Fig. 16 (bottom). When switching from N rq = 4 to N rq = 15, the C l errordecreases from 17 .
9% to 3 . c l,max c d,max FOM 0.0028 3.12ROM 0.0033 3.11Table 9: 3D flow past a cylinder: maximum lift and drag coefficients for FOM and ROM.Finally, we comment on the computational costs. The total CPU time required by aFOM simulation is 540 s. Our ROM approach takes 28 s. The speed-up is about 19, which ismuch smaller than the speed-up found for the 2D test. We identified two possible reasons: (i)the value of ¯ µ is smaller that one used for the 2D test and the high-fidelity solver convergesfaster (ii) the mesh used for the 3D test is coarser than one used for the 2D test. This work presents a POD-Galerkin based reduced order method for a Leray model im-plemented through the Evolve-Filter algorithm. Unlike the large majority of the works onLeray models, we choose a Finite Volume method because of its computational efficiency.The novelties of the proposed ROM approach are: (i) spatial filtering applied both for thecollection of the snapshots and in the reduced order model, (ii) the reconstruction of thepressure fields, and (iii) the use of different POD basis and coefficients to approximate thevelocity and pressure fields in the two steps of the Evolve-Filter algorithm. We assessed ourROM approach through two classical benchmarks: 2D and 3D flow past a cylinder. We foundthat our ROM can capture the flow features with an accuracy comparable to other ROMsapplied to similar benchmarks in [53, 55, 57]. In addition, we quantified the relative error inthe amplitude and phase of drag and lift coefficients computed by ROM and FOM. For the2D test case, we also performed a parametric study with respect to the filtering radius.A natural extension to the work presented in this manuscript is the development of aROM for a Leray model with a nonlinear differential filter. To this aim, we are working23igure 16: 3D flow past a cylinder: drag coefficient C d (top left) and lift coefficient C l (topright) computed by FOM and ROM for N ru = 9 and N rq = 4. A zoomed-in view of thelift coefficient around the time the maximum is reached is shown in the bottom row for twodifferent values of N rq . 24n order to extend the approach used in [30, 25], based on the idea of merging/combiningprojection-based techniques with data-driven reduction strategies. In particular, the strategyin [30, 25] exploits a data-driven reduction method to approximate the eddy viscosity solutionmanifold and a classical POD-Galerkin projection approach for the velocity and the pressurefields, respectively. We are going to use a data-driven reduction method to approximate theindicator function governing the amount of regularization introduced in the model in thenonlinear framework. We acknowledge the support provided by the European Research Council Executive Agencyby the Consolidator Grant project AROMA-CFD “Advanced Reduced Order Methods withApplications in Computational Fluid Dynamics” - GA 681447, H2020-ERC CoG 2015 AROMA-CFD, PI G. Rozza, and INdAM-GNCS 2019-2020 projects. This work was also partially sup-ported by US National Science Foundation through grant DMS-1620384 and DMS-195353.
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