Reduced-Order Modeling of Turbulent Flows ∗ J. Borggaard†, A. Duggleby‡, A. Hay§ , T. Iliescu¶, and Z. Wangk.
Abstract Proper orthogonal decomposition has been one of the main reducedorder modeling strategies in the study of turbulent flows. The model developed by Aubry et al. in [1] has been particularly successful in approximating the coherent structures in the turbulent boundary layer. For this model, we propose two improvements inspired from state-of-the-art ideas used in the large eddy simulation of turbulent flows: the variational multiscale [8] and dynamic subgrid-scale [6] methods. These developments are illustrated in the context of turbulent pipe flows. Key words reduced-order modeling, turbulence, variational multiscale
1
Introduction
Dynamical systems ideas have recently gained increased momentum in the study of turbulent flows. The reason is that dynamical systems could be used to describe low dimensional structures, such as the coherent structures in the turbulent boundary layer, which play an important role in the dynamics of the flow. ∗ We greatly appreciate the support of National Science Foundation via grants OCE 0620464, DMS 0209326, DMS 0513542, and the Air Force Office of Scientific Research via grants FA955005-1-0449 and FA9550-08-1-0136. † Department of Mathematics, Virginia Polytechnic Institute and State University, 528 McBryde Hall, Blacksburg, VA 24061, U.S.A., (
[email protected]). ‡ Department of Mechanical Engineering Texas A&M University, 109 Engineering/Physics Building Office Wing, College Station, TX 77843-3123, U.S.A., (
[email protected]). § Interdisciplinary Center of Applied Mathematics, Virginia Polytechnic Institute and State University, Wright House, Blacksburg, VA 24061, U.S.A., (
[email protected]). ¶ Department of Mathematics, Virginia Polytechnic Institute and State University, 456 McBryde Hall, Blacksburg, VA 24061, U.S.A., (
[email protected]). k Department of Mathematics, Virginia Polytechnic Institute and State University, 407E McBryde Hall, Blacksburg, VA 24061, U.S.A., (
[email protected]).
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One of the most successful dynamical systems ideas in the study of turbulent flows has been the Proper Orthogonal Decomposition (POD) [7]. POD starts with data from an accurate numerical simulation and then extracts the most energetic modes in the system by using the singular value decomposition. One of the main research areas in which POD has been used is to generate reduced-order models for the turbulent boundary layer. The first such model was proposed by Aubry et al. in [1]. This model has truncated the POD basis and has used an eddy viscosity type approximation to model the effect of the discarded POD modes on the POD modes kept in the model. The reduced-order model in [1] has yielded good qualitative results, considering the coarseness of the approximation. The criterion used to assess the accuracy of the POD model was the intermittency of the bursting events in the turbulent boundary layer. In two subsequent papers [10, 9], Podvin and Lumley have further investigated numerically the behavior of the POD model in [1]. They found that the model reproduced qualitatively well the physics of the turbulent boundary layer. Furthermore, by adding new POD modes to the model, the accuracy of the model was increased. The success of the reduced-order model in [1] relies fundamentally on the energy cascade assumption, which states that energy flows from low index POD modes to higher index POD modes. The validity of the extension of the energy cascade concept to the POD setting is the main focus of the paper of Couplet, Sagaut, and Basdevant [3]. The authors have investigated the energy transfer among POD modes in a non-homogeneous computational setting. By monitoring the triad interactions due to the nonlinear term in the Navier-Stokes equations, they have concluded that the transfer of energy among the POD modes is similar to the transfer of energy among Fourier modes. Specifically, they found that there is a net forward energy transfer from low index POD modes to higher index POD modes and that this transfer of energy is local in nature (that is, energy is mainly transfered among POD modes whose indices are close to one another). The studies in [1, 10, 9, 3] represent the motivation of our present study. These papers clearly suggest that the energy cascade concept is also valid in a POD setting. Therefore, Large Eddy Simulation (LES) ideas based on the energy cascade concept could also be used in devising reduced-order models in a POD context. Our current study proposes the use of the Variational Multiscale (VMS) method [8] and Dynamic Subgrid-Scale (DS) method [6] to improve the reduced-order POD model in [1]. The VMS approach, which is based on the concept of locality of energy transfer, and the DS approach, which computes dynamically the parameters in the POD model, have yielded highly successful LES models and appear as natural routes for reduced-order POD modeling. The rest of the paper is organized as follows. Section 2 describes the generation of DNS data for the turbulent pipe flow and section 3 briefly surveys the POD methodology. Section 4 derives several POD models based on successful LES ideas. Various challenges raised by the new POD models and the first steps in addressing them are illustrated numerically in section 5. Finally, section 6 presents conclusions and several research directions currently pursued by our group.
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2
Direct Numerical Simulation
The mathematical model for the flow of an incompressible, Newtonian fluid is given by the Navier-Stokes equations. Written in non-dimensional form, they read ∂t U + U · ∇U = −∇P + Re−1 τ ∆U in Ω × (0, T ],
(1)
∇·U=0
(2)
in ∂Ω × (0, T ],
where Ω is the computational domain, T the final time, U the velocity vector, Reτ the Reynolds number, P the pressure, andpt the time. The velocity is nondimensionalized by the wall shear velocity Uτ = τw /ρ, where τw is the wall shear stress and ρ is the density. The Reynolds number is Re τ = Uτ R/ν, where R is the radius of the pipe and ν is the kinematic viscosity. The method for solving these equations for all temporal and spatial scales is known as direct numerical simulation (DNS). The insight gained from the DNS of turbulence comes at a cost, with computational operations scaling as Re4τ and storage scaling as Re9/4 [2]. τ To illustrate numerically the two improvements we propose to the reducedorder POD model proposed in [1], we chose the turbulent pipe flow, which is a setting similar to that in [1]. To this end, a DNS of the turbulent pipe flow was carried out in [4]. The fully developed turbulent pipe flow that we consider is homogeneous in the azimuthal (θ) and streamwise directions, and is statistically steady. The DNS data are generated using a globally high-order spectral element algorithm, described in detail in [5]. The flow is driven by a mean streamwise pressure gradient to obtain a Reynolds number of Reτ = 150. When non-dimensionalized with the centerline velocity, the Reynolds number is Rec ≈ 4300. The domain length is L = 20R or z + = 3000 in wall units (distance non-dimensionalized by the length ν/Uτ ), which is long enough to simulate turbulent inlet and outlet boundary conditions with periodicity. Data are acquired for 2100 samples every t+ = 8. This corresponds to a total simulation time of t+ = Uτ2 t/ν = 16800 viscous time units, which is roughly 80 flow through times (tUm /L ≈ 80), where Um is the mean flow rate.
3
Proper Orthogonal Decomposition
The Proper Orthogonal Decomposition (POD) method provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of numerical experiments. POD provides the most efficient way of capturing the dominant components of an infinite-dimensional process with only finitely many, and often surprisingly few modes. We now briefly present the POD method in the context of turbulent pipe flow. For more details, the reader is referred to [7]. Let U(x, t), with (x, t) ∈ Ω × [0, T ], be the data obtained from the DNS of the turbulent pipe flow described in section 2. The POD functions are the spatial basis functions φ(x) that maximize h|(u, φ)|2 i, averaged projection of u onto φ, subject to the constraint kφk = 1. By using the Lagrange multipliers method, one can easily prove [7] that the optimal basis is given by the eigenfunctions of Z hU(x, t) ⊗ U(x0 , t)i φ(x0 ) dx0 = λ φ(x), (3) Ω
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where h·i denotes averaging. Using cylindrical coordinates for the turbulent pipe flow described in section 2 and considering centered data, the POD basis in (3) becomes the solution of Z
0
L Z 2π 0
Z
R
K(x, x0 ) φ(x0 ) r0 dr0 dθ0 dz 0 = λφ(x),
(4)
0
where K(x, x0 ) = hu(x, t) ⊗ u(x0 , t)i is the two-point velocity correlation kernel and u(x, t) = U(x, t) − U(x)
(5)
is the fluctuating component of the velocity U(x, t). The mean U(x) is determined by averaging in both homogeneous directions and time. The outer product ⊗ is used to represent the kernel as the two-point velocity correlation between every spatial point x = (r, θ, z) and x0 = (r0 , θ0 , z 0 ). For turbulent pipe flow, with two homogeneous directions providing translational invariance in θ (azimuthal) and z (streamwise) directions, we get K(x, x0 ) = K(r, r0 , θ − θ0 , z − z 0 ) = K(n, m; r, r0 ) einθ ei2πmz/L , where n is the azimuthal wavenumber and m is the streamwise wavenumber. Thus, the eigenfunctions have the form φn,m (r, θ, z) = ψ(n, m; r)einθ ei2πmz/L and ψ is the solution of
Z
(6)
R
K(n, m; r, r0 ) ψ(n, m; r0 ) r0 dr0 = λnm ψ(n, m; r), where 0
ˆ (n, m; r0 , t)i, λnm is the eigenvalue, and u ˆ (n, m; r) K(n, m; r, r0 ) = hˆ u(n, m; r, t) ⊗ u is the Fourier transform of the fluctuating velocity in the azimuthal and streamwise direction. Note that, since the flow data are solenoidal, so are φn,m . Solv-
Figure 1. Coherent vorticity of the φ151 (x) POD mode. ing this eigenvalue problem gives the POD decomposition of ψ(n, m; r). Since POD provides an orthogonal set of eigenfunctions that span the flow field, the method allows X Xthe Xflow field qto be represented as an expansion in that basis, u(x, t) = aqnm (t) φnm (x). For illustration purposes, we present the con
m
q
herent vorticity of the φ151 (x) POD mode in Fig. 1.
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4
POD Reduced-Order Models
In this section, we introduce the two improvements to the reduced-order POD model in [1]. To this end, we consider the following reduced-order approximation of the flow velocity U in (5) ur (x, t) = U(x) +
N X
M X
Q X
aqnm (t) φqnm (x),
(7)
n=−N m=−M q=1
where N is the number of Fourier modes in the azimuthal direction (θ), M is the number of Fourier modes in the streamwise direction (z), and Q is the number of POD functions in the radial direction (r). For clarity of notation, we will use the triplets k := (n, m, q). Thus, (7) takes place in the space (8) KN,M,Q := (n, m, q) |n| ≤ N, |m| ≤ M, 1 ≤ q ≤ Q .
We now replace U with ur in the Navier-Stokes equations (1)–(2), and then project the resulting equations onto the subspace span{φq }q∈KN,M,Q . Using the boundary conditions and the fact that all modes are solenoidal, one has: r 2 ∂u r r r , φk + ((u · ∇)u , φk ) + (u ), ∇φk = 0 ∀ k ∈ KN,M,Q , (9) ∂t Reτ
where (ur ) := (∇ur + (∇ur )T )/2 is the deformation tensor of ur . In order to ensure an accurate approximation of the reduced-order dynamics in (9), the effect of the discarded modes when replacing u with ur should be taken into account. Based on the similarity between POD and the closure problem in LES [7], different eddy-viscosity methodologies [1, 9] have been proposed to account for the effect of the discarded low-energy modes on the high-energy modes.
4.1
The mixing length POD model
The earliest attempt to model the effect of the discarded POD modes in (9) was made by Aubry at al. in [1]. This model is based on the mixing length hypothesis. The mixing length hypothesis is based on the concept of energy cascade [11], which states that energy enters the system at the largest scales, is transferred to smaller and smaller scales, and is dissipated at the smallest scales through viscous effects. Thus, the model for the effect of the discarded modes should be dissipative in the mean. The mixing length POD model in [1] reads r ∂u , φk + ((ur · ∇)ur , φk ) ∂t 2 2 νM L r (u ), ∇φk = 0 ∀ k ∈ KN,M,Q . (10) + + ν Reτ In (10), the following expression was proposed for νM L : νM L = α νT = α U L,
(11)
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where U and L are characteristic velocity and length scales for the unresolved scales, and α is a nondimensional parameter O(1) that characterizes the energy being dissipated. The parameter α is expected to vary in a real turbulent flow, and different values of α may result in different dynamics of the flow.
4.2
The Smagorinsky POD model
A potential improvement over the simplistic mixing length hypothesis is to replace νM L in (11) with a variable turbulent viscosity [12]. This can be achieved by using a Smagorinsky POD model : r ∂u , φk + ((ur · ∇)ur , φk ) ∂t 2 + νSmag + (ur ), ∇φk = 0 ∀ k ∈ KN,M,Q , (12) Reτ where νSmag := (CS L)2 | (ur )|,
(13)
CS is the Smagorinsky constant and | · | the Frobenius norm.
4.3
The Variational Multiscale POD model
The Variational Multiscale (VMS) methods were introduced by Hughes and his collaborators [8] and are based on the principle of locality of energy transfer: energy is transfered mainly between the neighboring scales. Couplet, Sagaut, and Basdevant [3] investigated the transfer of energy among POD modes for turbulent flow past a backward-facing step (a non-homogeneous separated flow). They discovered that the Fourier-decomposition based concepts of energy cascade and locality of energy transfer are also valid in the POD context (Figures 3 and 4 in [3]). This study provides the basis for our first development. We propose a new method to model the effect of the discarded POD modes in (9). We start by decomposing the finite set of POD modes KN,M,Q into the direct S sum of large resolved POD modes KL N,M,Q and small resolved POD modes KN,M,Q : S where (14) KN,M,Q = KL N,M,Q ⊕ KN,M,Q , L L L KL and (15) N,M,Q := (n, m, q) |n| ≤ N , |m| ≤ M , 1 ≤ q ≤ Q KSN,M,Q := (n, m, q) N L < |n| ≤ N, M L < |m| ≤ M, QL < q ≤ Q . (16)
Accordingly, we decompose ur into two components: urL representing the large resolved scales, and urS representing the small resolved scales: ur = urL + urS .
(17)
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The two components urL and urS represent the projections of ur onto the two spaces S KL N,M,Q and KN,M,Q , respectively. A decomposition for the pressure P is not necessary, since P does not appear in (9). The standard POD Galerkin approximation (9) can now be separated into two equations - one for urL (18) and one for urS (19). The Variational Multiscale POD model reads r ∂uL , φkL + (ur · ∇)ur , φkL ∂t 2 r ∀ k L ∈ KL (uL ), ∇φkL = 0 + N,M,Q , (18) Reτ r ∂uS , φkS + (ur · ∇)ur , φkS ∂t 2 r (uS ), ∇φkS = 0 ∀ kS ∈ KSN,M,Q , (19) + νSmag + Reτ where νSmag := (CS L)2 | (urS )|.
(20)
Note that equations (18) and (19) are coupled through the nonlinear term (ur ·∇)ur . The difference between the VMS POD model (18)–(19) and the Smagorinsky POD model (12) is that the former acts only on the small resolved scales, whereas the latter acts on all (both large and small) resolved scales.
4.4
Dynamic Subgrid-Scale POD model
Another avenue for improving the reduced-order model (9) is by computing the constant CS in the Smagorinsky POD model (20) dynamically, at each point in space and each time-step. This approach is inspired from the dynamic subgridscale procedure used in LES [6]. To this end, we will consider two filtering operations. In our POD context, the filtering operation is identified with the truncation of the POD modes used in the Galerkin method. We will denote the first POD filter by · and the second POD filter by e· . Note that modeling the effect of the discarded POD modes in (9) is equivalent to modeling the stress tensor [1] τi j := uri urj − ur i ur j .
(21)
We will also consider the following stress tensors: f f r ur − u r u r Ti j := u] i j i j
f f r ur − u r u r . and Si j := u^ i j i j
(22)
It is a simple calculation to check that the stress tensors τ := [τi j ], T := [Ti j ], and S := [Si j ] satisfy the following identity (called the “Germano identity” in LES): Si j = Ti j − τf i j.
(23)
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We assume the same eddy viscosity relation is defining the stress tensors τ and T : 1 1 e)| [ (u e)]ij . τij − τkk δij = −(CS L)2 | (u)| [ (u)]ij , Tij − Tkk δij = −(CS L)2 | (u 3 3 Equation (23) thus implies Si j −
1 2 Sk k δi j = CS2 γi j − C^ S βi j , 3
(24)
fr )| [ (u fr )] . We now where βi j := −L2 | (ur )| [ (ur )]i j and γi j := −L2 | (u ij 2 f 2 ^ assume that CS βi j = CS βi j , which is equivalent to assuming that CS2 remains unchanged within the secondary POD filtering e· . With this approximation, (24) 1 should be now implies that the residual Ei j := Si j − Sk k δi j − CS2 γi j − βf ij 3 identically zero, that is: S d − CS2 M = 0, where S d is the deviatoric part of the stress tensor S and the stress tensor M := [Mi j ] is defined as Mi j = γi j − βf i j . It is clear that this cannot be satisfied by a single constant CS . Thus, the following least squares problem is considered instead: d 2 d 2 min S − C M : S − C M . (25) S S 2 CS
Sd : M The solution CS2 to (25) is CS2 = . Since the stress tensors S d and M can be M :M computed directly from the resolved field, this yields a time- and space-dependent formula for CS (x, t) in (20). The dynamic subgrid-scale POD model reads r ∂u , φk + ((ur · ∇)ur , φk ) ∂t 2 + νDS + (ur ), ∇φk = 0 ∀ k ∈ KN,M,Q , (26) Reτ where νDS :=
Sd : M 2 L | (ur )|. M :M
(27)
Note that νDS defined in (27) can take negative values. This can be interpreted as backscatter, the inverse transfer of energy from high index POD modes to low index ones, which was found in [3].
5
Numerical results
To motivate the need for the improvements proposed in section 4, we illustrate the behavior of the 5-mode model proposed in Aubry et al. [1] in the numerical simulation of the entire turbulent pipe flow, and not only in the wall region of the turbulent boundary layer as in [1]. The POD decomposition of the turbulent pipe flow, as presented in section 3, leads to a spectrum for which the first five most energetic modes have no streamwise
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Index 1 2 3 4 5
n 6 5 3 4 2
m 0 0 0 0 0
q 1 1 1 1 1
Eigenvalue 1.61 1.48 1.45 1.29 1.26
Energy (% total) 2.42 % 2.22 % 2.17 % 1.93 % 1.88 %
Table 1. First 5 POD modes ranked in descending order of energy.
variations (m = 0) [1, 9]. These modes are given in Table 1 along with their corresponding eigenvalues and the percentage of the total energy of the fluctuating flow they represent. As can be seen, they account for less than 11% of the total energy of the fluctuating part of the flow. Note that the five modes corresponding to (−n, 0, 1) and their time histories are related to the previous ones for n > 0 from the fact that both the entries of the spatial correlation tensor and the flow are real : q∗ q ∀ (n, m, q) , φqnm (x) = φq∗ −n−m (x), and anm (t) = a−n−m (t), where the star sign denotes the complex conjugate. Thus, the resulting dynamical system is comprised of only five complex equations (10-dimensional model). We recall that the model also includes the mean flow through the centering trajectory. To account for this severe truncation, Aubry et al. have proposed in [1] the mixing length POD model (10). We consider α = νM L /νT (see equation (11)) as a bifurcation parameter for the present dynamical system. All reduced-order simulations are initialized from DNS data by projecting the spatial eigenfunctions on the first DNS snapshot.
0.002
0.01
1
1
Re(a30)
0.02
Re(a60)
0.004
0
-0.002
-0.004
0
-0.01
0
10000
20000
tU/R
(a) Re(a160 )
30000
40000
-0.02
0
10000
20000
30000
40000
tU/R
(b) Re(a130 )
Figure 2. Behavior of the 10-D model for α = 2.42.
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When α is chosen too large, the amount of dissipation introduced in the model is too high, and thus the global attractor of the system is the trivial solution where all time coefficients tend to zero. This behavior is illustrated in Fig. 2 for α = 2.42. The modes are not allowed to exhibit any dynamics since the model systematically produces too much dissipation in the model. The opposite effect is observed when α is chosen too small as illustrated in Fig. 3 for α = 2.41. In this case, the model does not dissipate enough energy so that the solution grows in time without bound. For this value of the eddy viscosity, the model underpredicts the amount of energy that the resolved modes should send to the neglected scales.
0.08
0.08
0.04
0.04 1
1
Re(a30)
0.12
Re(a60)
0.12
0
0
-0.04
-0.04
-0.08
-0.08
-0.12
0
10000
20000
tU/R
(a) Re(a160 )
30000
40000
-0.12
0
10000
20000
30000
40000
tU/R
(b) Re(a130 )
Figure 3. Behavior of the 10-D model for α = 2.41. We also varied the parameter α to investigate whether the low-dimensional model can exhibit the intermittent features reminiscent of the bursting events observed in a full-order wall-bounded turbulent flow. Such a result from a lowdimensional model was first reported in [1]. This was one of the main findings of [1], since the low dimensionality of the model allows for the understanding and possibly control of the main dynamics of the flow. In our study, however, we were not able to observe any intermittent features. The reason is probably due to the significant differences between our setting and that in [1]. An important difference is that in [1] only the wall region of the turbulent boundary layer was considered. In this study, we used a reduced-order model with the same number of POD modes as the model in [1] for the entire turbulent pipe flow. Equally important, we are not using any feedback to provide the necessary adaptivity in time from modeling the mean flow. This adaptivity is considered essential in [1, 7]. Finally, some parameters used in our study (such as Reτ ) are different from those used in [1].
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6
Conclusions
In this study, we have developed two improvements over the reduced-order POD model of Aubry et al. [1]. To this end, we have adapted state-of-the-art ideas used in LES of turbulent flows to the POD context. Specifically, we have used the variational multiscale and dynamic subgrid-scale methods proposed in [8] and [6], respectively. These developments are presented and illustrated numerically in the context of turbulent pipe flows, similar to the setting used in [1]. We only made the first steps in testing numerically these new developments. Specifically, we tried the model developed in [1] in our computational setting. There are, however, several differences between our study and that in [1]. The most important difference is that this model was used in [1] for the approximation of coherent structures in the wall region of the turbulent boundary layer, whereas we are trying to approximate the entire pipe flow. Moreover, we are not using any feedback to provide the necessary adaptivity in time from modeling the mean flow [1, 7]. Finally, the parameters used in our study (such as Reτ ) are different from those used in [1]. Based on these first numerical results, we drew the following conclusions. First, it is apparent that the reduced-order POD model needs to include more POD modes in order to approximate accurately the entire pipe flow, and not only the turbulent boundary layer as in [1]. Second, even with an increased number of POD modes, the reduced-order POD model equipped with the simple mixinglength model to account for the POD modes not included in the model, will probably still perform poorly. Adapting state-of-the-art large eddy simulation modeling methodologies could improve significantly the performance of the reduced-order POD model. Finally, the criterion used in [1] to evaluate the performance of the reduced-order POD model is its ability to reproduce the intermittency of the coherent structures in the boundary layer. For a truly successful reduced-order POD model, other criteria of interest in turbulent flow simulation and control should be satisfied. We plan to investigate all these research directions in a future study.
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