Using POD and DMD for comparing CFD and experimental results in unsteady aerodynamics

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Using POD and DMD

for comparing CFD and experimental results

in unsteady aerodynamics

A. Guissart, T. Andrianne,

G. Dimitriadis & V.E. Terrapon

University of Liege

November 25, 2014

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Motivation

Comparison of results

Experiments (EFD) Simulations (CFD)

3 Global quantities

7 Local quantities

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Motivation

Comparison of results

Experiments (EFD) Simulations (CFD)

3 Global quantities

7 Local quantities

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POD and DMD used for comparison

POD finds the most persistent spatial structures φ

POD i

u px, y , tq “

N

ÿ

i “1

a

POD i

lo

omo

on

amplitude = energy

f

POD i

ptq

loomoon

time evolution

φ

POD i

px , y q

looooomooooon

spatial mode

DMD finds single frequency modes φ

DMD i

u px, y , tq “

N

ÿ

i “1

a

DMD i

lo

omo

on

amplitude

exp pλ

DMD i

tq

looooomooooon

time evolution

φ

DMD i

px , y q

looooomooooon

spatial mode

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POD and DMD used for comparison

POD finds the most persistent spatial structures φ

POD i

u px, y , tq “

N

ÿ

i “1

a

POD i

lo

omo

on

amplitude = energy

f

POD i

ptq

loomoon

time evolution

φ

POD i

px , y q

looooomooooon

spatial mode

DMD finds single frequency modes φ

DMD i

u px, y , tq “

N

ÿ

i “1

a

DMD i

lo

omo

on

amplitude

exp pλ

DMD i

tq

looooomooooon

time evolution

φ

DMD i

px , y q

looooomooooon

spatial mode

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Methodology

Data Collection Decomposition Comparison

Static rectangle

‚

Re “ Uc{ν “ 105

‚

c “ 4d

‚

α “ 50 Pitching rectangle

‚

Re “ 4 ¨ 104

‚

k “ πfd{U “ 6.85 ¨ 10´2

‚

αMAX “ 100

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Methodology

Data Collection Decomposition Comparison

Static rectangle

‚

Re “ Uc{ν “ 105

‚

c “ 4d

‚

α “ 50 Pitching rectangle

‚

Re “ 4 ¨ 104

‚

k “ πfd{U “ 6.85 ¨ 10´2

‚

αMAX “ 100

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Methodology

Data Collection

1. EFD Ñ Tr-PIV

2. CFD Ñ uRANS

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Methodology

Data Collection 1. EFD Ñ Tr-PIV 2. CFD Ñ uRANS Decomposition 1. POD: φipx , y q and ai 2. DMD: φipx , y q, ai and λi

a

DMD i

exp

`λ

DMD_i

t

˘

φ

DMD i

px , y q

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Methodology

Data Collection 1. EFD Ñ Tr-PIV 2. CFD Ñ uRANS Decomposition 1. POD: φipx , y q and ai 2. DMD: φipx , y q, ai and λi Comparison 1. φipx , y q 2. ai and λi u px, y , tq “ N ÿ i “1 aDMD i loomoon amplitude exp`λDMD i t ˘ looooomooooon time evolution φDMDi px , y q looooomooooon spatial mode

Modes shapes comparison MAC´φEFD_i , φCFD_j ¯“ ¨ ˝ φEFD_i ¨ φCFD_j }φEFD i }}φCFDj } ˛ ‚ 2

Amplitudes and frequencies comparison ERR´aEFD_i , aCFD_i ¯“}a

EFD

i } ´ }aCFDi } }aEFD

i }

Modal Assurance Criterion

MAC ´ φEFDi , φCFDj ¯ “ ˜ φEFD i ¨ φCFDj }φEFDi }}φCFDj } ¸2 1 2 3 4 1 2 3 4 CFD modes EFD mo des 0 0.2 0.4 0.6 0.8 1

MAC takes values between 0 and 1

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Methodology

Data Collection 1. EFD Ñ Tr-PIV 2. CFD Ñ uRANS Decomposition 1. POD: φipx , y q and ai 2. DMD: φipx , y q, ai and λi Comparison 1. φipx , y q 2. ai and λi u px, y , tq “ N ÿ i “1 aDMD i loomoon amplitude exp`λDMD i t ˘ looooomooooon time evolution φDMDi px , y q looooomooooon spatial mode

Modes shapes comparison MAC´φEFD_i , φCFD_j ¯“ ¨ ˝ φEFD_i ¨ φCFD_j }φEFD i }}φCFDj } ˛ ‚ 2

Amplitudes and frequencies comparison ERR´aEFD_i , aCFD_i ¯“}a

EFD i } ´ }aCFDi }

}aEFD

i }

Modal Assurance Criterion

MAC ´ φEFDi , φCFDj ¯ “ ˜ φEFD i ¨ φCFDj }φEFDi }}φCFDj } ¸2 1 2 3 4 1 2 3 4 CFD modes EFD mo des 0 0.2 0.4 0.6 0.8 1

MAC takes values between 0 and 1

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Oscillating rectangle

‚

EFD: rectangle undergoing LCO

‚

CFD: pitching frequency and

amplitude imposed (same as EFD)

Problem

‚

No other EFD values than velocities

‚

CFD data need to be validated

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Oscillating rectangle

POD results Energy content 5 10 15 20 10−2 10−1 100 101 102 Mode Con tribution to total energy (%) EFD CFD MAC matrix 1 2 3 4 5 6 1 2 3 4 5 6 0.99 0.86 0.76 0.52 0.19 0.19 CFD modes EFD mo des 0 0.2 0.4 0.6 0.8 1

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Static rectangle

Global values cl cd St CFD 0.83 0.46 0.136 EFD 0.53 0.45 0.152 Problem

‚

cl from CFD far from EFD value

‚

Why?

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Static rectangle

DMD results EFD: 2 modes CFD: 2 modes EFD: 3 modes In CFD reconstruction

‚

More important reverse flow

‚

Discrepancies on the rear part

‚

Different dynamic of reattachment

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Static rectangle

‚

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Static rectangle

‚

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Static rectangle

‚

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Conclusion and future work

DMD and POD are useful

‚

Compare and validate CFD results

‚

Highlight and understand potential

discrepancies

‚

Enlarge PIV window to get the rear part

‚

Apply DMD on lift evolution

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Static rectangle

POD results Energy content 2 4 6 8 10 10−5 10−3 10−1 101 Mode Con tribution to total energy (%) EFD CFD MAC matrix 1 2 3 4 1 2 3 4 0.98 0.41 0.38 0.2 0.15 0.19 CFD modes EFD mo des 0 0.2 0.4 0.6 0.8 1