Dissipation-optimized Proper Orthogonal Decomposition P. J. Olesen1A. Hodžić1S. J. Andersen2N. N. Sørensen3and C. M. Velte1 1Department of Civil and Mechanical Engineering Technical University of Denmark 2800 Kgs. Lyngby

2025-05-03 0 0 3.21MB 16 页 10玖币

侵权投诉

Dissipation-optimized Proper Orthogonal Decomposition

P. J. Olesen,1A. Hodžić,1S. J. Andersen,2N. N. Sørensen,3and C. M. Velte1

1)Department of Civil and Mechanical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby,

Denmark

2)Department of Wind and Energy Systems, Technical University of Denmark, 2800 Kgs. Lyngby,

Denmark

3)Department of Wind and Energy Systems, Technical University of Denmark, 4000 Roskilde,

Denmark

(*Electronic mail: pjool@dtu.dk)

(Dated: 26 October 2022)

We present a formalism for dissipation-optimized decomposition of the strain rate tensor (SRT) of turbulent ﬂow data

using Proper Orthogonal Decomposition (POD). The formalism includes a novel inverse spectral SRT operator allow-

ing the mapping of the resulting SRT modes to corresponding velocity ﬁelds, which enables a complete dissipation-

optimized reconstruction of the velocity ﬁeld. Flow data snapshots are obtained from a direct numerical simulation of

a turbulent channel ﬂow with friction Reynolds number Reτ=390. The lowest dissipation-optimized POD (d-POD)

modes are compared to the lowest conventional turbulent kinetic energy (TKE) optimized POD (e-POD) modes. The

lowest d-POD modes show a richer small-scale structure, along with traces of the large-scale structure characteristic

of e-POD modes, indicating that the former capture structures across a wider range of spatial scales. Proﬁles of both

TKE and dissipation are reconstructed using both decompositions, and reconstruction convergences are compared in all

cases. Both TKE and dissipation are reconstructed more efﬁciently in the dissipation-rich near-wall region using d-POD

modes, and in the TKE-rich bulk using e-POD modes. Lower modes of either decomposition tend to contribute more

to either reconstructed quantity. Separating each term into eigenvalues and factors relating to the inherent structures in

each mode reveals that higher e-POD modes tend to encode more dissipative structures, whereas the structures encoded

by d-POD modes have roughly constant inherent TKE content, supporting the hypothesis that structures encoded by

d-POD modes tend to span a wide range of spatial scales.

I. INTRODUCTION

Viscous dissipation is a parameter crucial for understanding

and modeling dynamics of turbulent ﬂows (Pope, 2000). Ex-

isting approaches to ﬂow decompositions used e.g. for gaining

insight into ﬂow dynamics and for formulating reduced or-

der models (ROMs) typically educe structures based on their

turbulent kinetic energy (TKE) content, neglecting dissipa-

tive structures. Various techniques have been developed to

compensate for unresolved dissipative structures in ROMs;

however, explicitly optimizing the ﬂow decomposition with

respect to dissipation would enable closer studies of dissipa-

tive structures, while also providing an avenue for construct-

ing more robust ROMs.

The application of proper orthogonal decomposition (POD)

to turbulent ﬂows was pioneered by Lumley (1967) with the

aim of objectively identifying and characterizing dominant

large scale ﬂow structures. Applying this classical POD to

a velocity ﬂuctuation ﬁeld ensemble produces an orthogonal

basis spanning the ensemble such that reconstructing the ﬂuc-

tuation ﬁeld using a given number of modes reproduces the

optimal amount of mean TKE compared to any other decom-

position. In this sense, a POD-based modal expansion is op-

timally robust against truncation (see Berkooz, Holmes, and

Lumley, 1993; Holmes et al., 2012). This explicit optimiza-

tion with respect to TKE is what leads to large-scale structures

being prioritized in the decomposition.

Gatski and Glauser (1992) applied the classical POD to

channel ﬂow data obtained from a Direct Numerical Simu-

lation (DNS) and reconstructed the TKE, the shear stresses,

and the dissipation, demonstrating that while the former two

could be reconstructed accurately using relatively few modes,

the reconstruction of the dissipation required a larger num-

ber of modes. This was taken as a conﬁrmation of the work

of Ukeiley et al. (1992), which suggested that the ordering of

POD modes by decreasing mean TKE contribution was equiv-

alent to ordering by decreasing characteristic length scales in

the ﬂow structures described by the modes. Dissipation, be-

ing related to small scale motions, would therefore be poorly

reconstructed using an expansion favoring large scales. The

parallel between POD modes and length scales was extended

by Couplet, Sagaut, and Basdevant (2003), who showed that

for a ﬂow past a backward-facing step, the concept of the en-

ergy cascade from larger to smaller length scales described by

Richardson (1922) also applied to POD modes in this ﬂow;

their analysis of triadic terms in the Galerkin-projected energy

transport equation showed a local net ﬂow of TKE from lower

towards higher POD modes. The implication, in agreement

with the ﬁndings of Gatski and Glauser (1992), is that higher

POD modes are needed to describe the small-scale velocity

ﬂuctuations.

Ali, Kadum, and Cal (2016) applied multifractal analysis

to wind turbine wake dissipation signals reconstructed using

either the lowest several POD modes, associated with larger

length scales, or the remaining modes, associated with smaller

length scales. It was demonstrated that the multifractal struc-

ture of the full dissipation signal was reproduced more accu-

rately using higher modes than using lower modes, supporting

the notion that dissipation information is encoded in higher

modes. On the other hand, it was shown by Lee and Dowell

(2020) that for an anisotopic 2D ﬂow, reordering POD modes

arXiv:2210.12533v2 [physics.flu-dyn] 25 Oct 2022

by their mean velocity gradient norm contribution rather than

their mean TKE contribution resulted in only a small degree

of rearrangement. This was interpreted as evidence against

modal scale ordering and modal cascade, and it was suggested

instead that each POD mode captures a wide range of dynam-

ical scales; however, this result must be viewed in the light of

the different dynamics in play for 2D turbulence compared to

the 3D case.

The association of small-scale structures with higher and

less energetic POD modes suggests that modal optimality with

respect to TKE causes small-scale ﬂuctuations to be under-

represented: a representation that prioritizes TKE-rich larger

scales will be inefﬁcient at reconstructing the smaller scales

characterizing dissipation. This trade-off means that POD-

based ROMs lack accurate representation of small-scale ﬂuc-

tuations important in determining dissipation, which is a cen-

tral parameter in turbulence theory and modeling. This under-

representation of small scales has often been associated with

model instability (see Bergmann, Bruneau, and Iollo, 2009),

although this claim has been challenged by Grimberg, Farhat,

and Youkilis (2020) who demonstrated that instabilities are

inherent to the Galerkin formalism rather than a consequence

of the scales represented in the basis. This notwithstanding,

model accuracy relies on the choice of basis: since turbulent

dynamics are characterized by interactions between a wide

range of scales, even low energetic structures generally play

an important role in determining the dynamics of the ﬂow.

To achieve accurate POD-based ROMs one may there-

fore include corrections to account for the neglected modes.

Bergmann, Bruneau, and Iollo (2009) presented an overview

of approaches to address this issue, of which only a few will

brieﬂy be recounted here. In the earliest work on POD-based

ROMs, Aubry et al. (1988) used a generalized Heisenberg

model in which the effective viscosity was adjusted to cor-

rect for unresolved modes. Iollo et al. (2000) described two

avenues to stabilize ROMs, namely compensation for unre-

solved modes by the addition of an explicit dissipation term,

and inclusion of the velocity gradient in the norm with re-

spect to which optimization was performed, though the lat-

ter approach was not realized in that work. Furthermore, Lee

and Dowell (2020) formulated a 2D ROM in terms of two

POD bases, supplementing the TKE-optimized velocity ﬂuc-

tuation ﬁeld representation with an enstrophy-optimized gra-

dient ﬁeld representation. Andersen and Murcia Leon (2022)

showed how stability can be achieved by ensuring that the

non-linear interactions across modes are correctly maintained

when utilizing a global POD basis.

In the present work we build upon the concept presented by

Lee and Dowell (2020), introducing an explicitly dissipation-

optimizing POD formulation (d-POD). The resulting d-POD

modes span the set of strain rate tensors (SRTs) derived from

the ensemble of velocity ﬂuctuation ﬁelds. By introducing an

inverse spectral SRT operator the modes can be mapped to ve-

locity ﬁelds, facilitating their use as a supplement to conven-

tional TKE-optimized (e-POD) modes. We show that d-POD

modes computed from cross sectional slabs of a 3D channel

ﬂow DNS (Reτ=390) reconstruct the dissipation proﬁle more

efﬁciently than do e-POD modes, and vice versa for the TKE

proﬁle.

Although this is to our knowledge the ﬁrst instance of an

explicitly dissipation-optimized POD, several works, in addi-

tion to Lee and Dowell (2020), have employed optimization

with respect to the closely related enstrophy. While distinct

quantities, as discussed e.g. by Bermejo-Moreno, Pullin, and

Horiuti (2009) and Yeung, Donzis, and Sreenivasan (2012)

dissipation and enstrophy remain intimately coupled and dis-

play similar spectral properties. They are proportional to

the squared norm of the SRT and the vorticity, respectively,

both of which are constructed from ﬁrst-order gradient terms

∇jui. Similar properties with respect to POD convergence

may therefore be expected, providing the basis for comparison

between results found in the present work and comparable re-

sults on enstrophy-based POD found in literature. Enstrophy-

based POD was utilized by Huang (1994) and by Kostas, So-

ria, and Chong (2005) for identifying coherent ﬂow structures

from 2D particle image velocimetry (PIV) measurements of

a backward-facing step ﬂow, and similarly by Munir et al.

(2022), based on 2D measurements of two-phase slug ﬂow

made by combining PIV and laser-induced ﬂuorescence. Sen-

gupta et al. (2015) demonstrated reduced-order modeling of

a ﬂow past a cylinder applying enstrophy-based POD to 2D

DNS data, using the resulting modes in a vorticity-formulation

of the Navier-Stokes equations. Like conventional e-POD

modes, enstrophy-based POD modes allow for identiﬁcation

and characterization of important ﬂow structures; the method

is of particular use in inhomogeneous ﬂows where vortical

structures are of central importance to the dynamics (see Sen-

gupta, 2012).

The remainder of this paper is laid out as follows. In section

II we describe the the general POD formalism and its adap-

tations in the form of TKE and dissipation optimized PODs

employed in this work, we introduce the inverse spectral SRT

operator, and we brieﬂy discuss the reconstruction of TKE

and dissipation based on the above. Implementation details,

including the DNS study used to produce the data on which

the analysis is built, are discussed in section III. We present a

comparison of the lowest POD modes in section IV. TKE and

dissipation are each reconstructed using either basis set, and

we analyze the reconstruction of proﬁles and total quantities

in section V. Discussion and conclusions are given in sections

VI and VII, respectively.

II. PROPER ORTHOGONAL DECOMPOSITION

The POD provides a decomposition of an ensemble of vec-

tors that is optimally efﬁcient as measured by the norm on the

underlying vector space (see e.g. Holmes et al., 2012; Weiss,

2019). A formal and rather generic description of POD is

given in section II A. The speciﬁcations of the formalism rel-

evant to the energy and dissipation based PODs will be dis-

cussed in section II B. In section II C we introduce the inverse

spectral SRT operator used to map d-POD modes to velocity

ﬁelds. The modal reconstruction of TKE and dissipation is

discussed in section II D.

A. Generic POD Formalism

Let Hbe a real Hilbert space of dimension Nwith inner

product (·,·)H:H×H→R, and let α∈Hbe an arbitrary

element in H. The induced norm k·kH:H→R0+is given

by kαkH=|(α,α)H|1

2. Let F={fm}M

m=1⊂Hbe a set of

Msamples from H.

Using the averaging operation h·i, we deﬁne the POD oper-

ator R:H→Hfor Fby its action on α,

Rα=D{(α,fm)Hfm}M

m=1E.(1)

The operator Ris Hermitian and has orthogonal eigenvec-

tors {φn}N

n=1and real and non-negative eigenvalues {λn}N

n=1,

Rφn=λnφn,λn≥0,(φn,φn0)H=δnn0,(2)

where δnn0is the Kronecker delta. The eigenvectors are the

POD modes, the set of which forms the POD basis, a com-

plete orthogonal basis for F. By convention the POD ba-

sis is ordered by decreasing eigenvalues such that λ1≥λ2≥

... ≥λN≥0, and the POD modes are normalized, kφnkH=1

for n=1,2,...N. The number of non-zero eigenvalues is

rank(F)≤min(M,N); in most applications MN, and M

eigenpairs will be sufﬁcient to completely span F. For the

sake of generality, we will maintain Nas our notation for the

number of eigenpairs, although in implementations we shall

set N=M−1 since the sample mean is subtracted from each

sample, reducing the sample set rank by one.

Each sample fm∈Fcan be expanded in the POD basis

using coefﬁcients {amn}N

n=1,

fm=

∑

n=1

amnφn,amn = (φn,fm)H.(3)

These coefﬁcients are uncorrelated, satisfying

D{amnamn0}M

m=1E=λnδnn0.(4)

The optimality of the POD basis with respect to the inner

product on Hcan be stated formally as

φn=argmax

φ∈H

n|(φ,fm)H|2oM

m=1

kφk2

for n=1,2,...,N.

(5)

The eigenvalue problem (2) is derived from this optimization

problem (Lumley, 1967). It results in an optimal basis in the

following sense. A sample fm∈Fmay be expanded using

the POD basis {ϕn}N

n=1, or alternatively using an arbitrary or-

thogonal basis {ϕ0

n}N

n=1spanning the same subspace of Has

the POD basis. Given ˆ

N≤Nthe sample fmmay then be ap-

proximated by its truncated expansion in either basis, {φn}ˆ

n=1

or {φ0

n}ˆ

n=1:

fm≈ˆ

fm=

∑

n=1

amnφn;fm≈ˆ

∑

n=1

mnφ0

n.(6)

The POD basis minimizes the mean squared error of the ap-

proximation as measured by the norm on H, compared to an

arbitrary orthogonal basis,

Dkˆ

fm−fmk2

HM

m=1E≤Dkˆ

m−fmk2

HM

m=1E,(7)

for any ˆ

N≤N. The sense in which the POD basis is optimal is

thus fully determined by the choice of norm k·kH, and hence

by the choice of Hilbert space and inner product.

B. Energy and dissipation based PODs

We construct two different POD bases using the general

formalism laid out in the previous section. One is the con-

ventional TKE-based POD (e-POD), which uses a sample set

U={um}M

m=1⊂Heof velocity ﬂuctuation snapshots on the

spatial domain Ωe, where the Hilbert space Heis

He:=(α:Ωe→R3

∑

i=1ZΩeαiαidx <∞).(8)

The inner product on He,(·,·)He:He×He→Ris deﬁned

(α,β)He=

∑

i=1ZΩeαiβidx .(9)

The POD basis {ϕn}N

n=1, resulting from solving (2), is optimal

with respect to the mean turbulent kinetic energy (TKE) of the

ﬂow, and the associated eigenvalues are {λe

n}N

n=1. The mean

TKE density, hTi, and total mean TKE, T, are

hTi=1

2n|um|2oM

m=1=1

∑

n=1

λe

n|ϕn|2,(10a)

T=ZΩhTidx =1

∑

n=1

λe

n,(10b)

where |α|2=∑3

i=1αiαifor α∈He. This is the commonly

used space-only POD, which allows a TKE-optimal recon-

struction of the velocity ﬁeld,

um=

∑

n=1

amnϕn,amn = (ϕn,um)He.(11)

We propose a second POD variant, namely the dissipation-

based POD (d-POD), using instead the ensemble of SRTs

computed from the original snapshot ensemble U, i.e., S=

{si j

m}M

m=1⊂Hdand deﬁned on the spatial domain Ωd. The

Hilbert space Hdis

Hd:=(α:Ωd→R3×3

∑

i,j=1ZΩdαi jαi j dx <∞),(12)

with the inner product (·,·)Hd:Hd×Hd→Rdeﬁned as

(α,β)Hd=

∑

i,j=1ZΩdαi jβi j dx .(13)

Assuming differentiability of each component of each ve-

locity ﬂuctuation snapshot, um, the components of the corre-

sponding strain rate tensor, sm, are obtained as si j

m≡(Dum)i j,

with the SRT operator D:He→Hdgiven by

(Dα)i j =1

2∇jαi+∇iαj.(14)

The resulting d-POD modes, {ψn}N

n=1, with associated

eigenvalues, {λd

n}N

n=1, provide a dissipation-optimal recon-

struction of the SRT ﬁeld,

sm=

∑

n=1

bmnψn,bmn = (ψn,sm)Hd.(15)

The mean norm on Sis proportional to the mean viscous

dissipation of the ﬂow. The mean dissipation density hεiand

the mean total dissipation Eare given by

hεi=2νn|sm|2oM

m=1=2ν

∑

n=1

λd

n|ψn|2,(16a)

E=ZΩhεidx =2ν

∑

n=1

λd

n,(16b)

where |α|2=∑3

i,j=1αi jαi j for α∈Hd.

C. The spectral inverse SRT operator

An inverse mapping exists, Hd→He, mapping SRTs to

velocity ﬂuctuation ﬁelds. To establish an explicit spectral

form of this mapping, D−1, we consider the subset of d-POD

modes for which λd

n>0, and insert the POD operator deﬁni-

tion from (1) using Sin the eigenvalue problem (2),

Rψn=D{(ψn,sm)Hdsm}M

m=1E=λd

nψn.(17)

Then apply D−1to ψnusing D−1sm=umto yield

D−1ψn=D{(ψn,sm)Hdum}M

m=1E

λd

=D{bmnum}M

m=1E

λd

(18)

The set {D−1ψn}λd

n>0spans the velocity sample space U,

um=∑

n|λd

n>0

bmnD−1ψn,(19)

allowing the reconstruction of any velocity-dependent quan-

tity in terms of dissipation-optimized modes. While

{D−1ψn}λd

n>0does form a complete basis for U, we note that

the basis is in general not orthogonal with respect to the inner

product (·,·)He. It is, however, orthogonal (and normalized)

with respect to the inner product (·,·)H0deﬁned as

(·,·)H0= (D·,D·)Hd,(20)

since (D−1ψn,D−1ψn0)H0= (ψn,ψn0)Hd=δnn0. The opera-

tor Din (14) and its inverse D−1in (18) thus link the spaces

of velocity ﬂuctuation ﬁelds and of SRT ﬁelds to each other,

as shown schematically in Figure 1; this link provides useful

ﬂexibility when working with the two bases.

{um}M

m=1{sm}M

m=1

nϕn,λe

n,{amn}M

m=1oN

n=1nψn,λd

n,{bmn}M

m=1oN

n=1

D−1ψnλd

n>0{Dϕn}N

n=1

HeHd

e-POD d-POD

D−1

FIG. 1. Schematic depiction of the relation between the objects ap-

pearing in the POD. Objects on the left side are associated with the

space of velocity ﬁelds (He), while those on the right are associ-

ated with the space of strain rate tensor ﬁelds (Hd). The upper row

represents samples, the middle row the POD results (modes, eigen-

values, and coefﬁcients), and the lower row the velocity ﬂuctuation

ﬁelds computed from d-POD modes and the strain rate tensor ﬁelds

computed from e-POD modes.

D. Modal reconstructions of TKE and dissipation

By applying the spectral inverse SRT operator the mean

TKE and dissipation densities can be reconstructed using ei-

ther basis. We wish to compare the efﬁciency of these recon-

structions in terms of their rate of convergence, leading us to

consider the expansions truncated to include ˆ

N≤Nmodes.

The mean TKE density ﬁeld reconstructed using ˆ

Ne-POD or

d-POD modes is

DTe,ˆ

NE=1

∑

n=1

λe

n|ϕn|2,(21a)

DTd,ˆ

NE=1

∑

n=1

λd

nD−1ψn2,(21b)

and the mean dissipation density is likewise reconstructed by

Dεe,ˆ

NE=2ν

∑

n=1

λe

n|Dϕn|2,(22a)

Dεd,ˆ

NE=2ν

∑

n=1

λd

n|ψn|2.(22b)

The e-POD and d-POD reconstructions agree for each

quantity when ˆ

N=N, but generally exhibit different conver-

gence rates.

III. CHANNEL FLOW

To demonstrate the method presented above we apply it to

an ensemble of turbulence ﬂuctuation velocity data {um}M

m=1

obtained from a DNS of a double-periodic channel ﬂow. This

is chosen as an example of a relatively simple and well-

understood ﬂow, while still exhibiting shear layers and inho-

mogeneous turbulence which highlight relevant features of the

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Dissipation-optimizedProperOrthogonalDecompositionP.J.Olesen,1A.Hodºi¢,1S.J.Andersen,2N.N.Sørensen,3andC.M.Velte11)DepartmentofCivilandMechanicalEngineering,TechnicalUniversityofDenmark,2800Kgs.Lyngby,Denmark2)DepartmentofWindandEnergySystems,TechnicalUniversityofDenmark,2800Kgs.Lyngby,Denmark3)Depa...

展开>> 收起<<

Dissipation-optimized Proper Orthogonal Decomposition P. J. Olesen1A. Hodžić1S. J. Andersen2N. N. Sørensen3and C. M. Velte1 1Department of Civil and Mechanical Engineering Technical University of Denmark 2800 Kgs. Lyngby.pdf

共16页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Dissipation-optimized Proper Orthogonal Decomposition P. J. Olesen1A. Hodžić1S. J. Andersen2N. N. Sørensen3and C. M. Velte1 1Department of Civil and Mechanical Engineering Technical University of Denmark 2800 Kgs. Lyngby

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: