Completion of partially known second-order statistics of turbulent - - PowerPoint PPT Presentation

▶
completion of partially known second order statistics of
SMART_READER_LITE
LIVE PREVIEW

Completion of partially known second-order statistics of turbulent - - PowerPoint PPT Presentation

Jun 18, 2023 134 likes •323 views

Completion of partially known second-order statistics of turbulent flows Armin Zare Joint work with: Mihailo R. Jovanovi c Tryphon T. Georgiou 67th APS DFD Annual Meeting, San Francisco, 2014 1 / 17 Low-complexity models of turbulent

slide-1
SLIDE 1

Completion of partially known second-order statistics

  • f turbulent flows

Armin Zare

Joint work with: Mihailo R. Jovanovi´ c Tryphon T. Georgiou

67th APS DFD Annual Meeting, San Francisco, 2014

1 / 17

slide-2
SLIDE 2

Low-complexity models of turbulent flows

I Control-oriented modeling

  • Stochastically forced NS equations

ψt = A ψ + f v = C ψ

I Motivation

  • linear mechanisms in self-sustaining cycle of near-wall structures

Hamilton, Kim, Waleffe, JFM ’95

  • forcing statistics influence performance of flow estimators

Chevalier, Hœpffner, Bewley, Henningson, JFM ’06

2 / 17

slide-3
SLIDE 3

Proposed Approach

I view second-order statistics as data for an inverse problem I identify forcing statistics to account for partially known statistics

3 / 17

slide-4
SLIDE 4

White-in-time stochastic forcing

I State statistics - white-in-time u

Lyapunov equation

  • !

A X + X A∗ = B WB∗

I Success in capturing qualitative features Farrell & Ioannou, POF ’93 Bamieh & Dahleh, POF ’01 Jovanovi´ c & Bamieh, JFM ’05 Moarref & Jovanovi´ c, JFM ’12

4 / 17

slide-5
SLIDE 5

I Key observation

λi (A XDNS + XDNS A∗)

  • White-in-time stochastic excitation too restrictive!

Jovanovi´ c & Georgiou, APS ’10

5 / 17

slide-6
SLIDE 6

Colored-in-time stochastic forcing

I State statistics - colored-in-time u

A X + X A∗ = (B H∗ + H B∗) | {z } Q

Georgiou, IEEE TAC ’02 Chen, Jovanovi´ c, Georgiou, CDC ’13

6 / 17

slide-7
SLIDE 7

An example

I Response of a boundary layer to free-stream turbulence

7 / 17

slide-8
SLIDE 8

Inverse problem

I Covariance completion problem

minimize

Q, X

kQk∗ subject to A X + X A∗ + Q = 0 trace ( Tk X ) = gk, k = 1, . . . , N X ⌫ 0

I kQk∗ :=

X σi(Q)

  • !

proxy for rank minimization

Fazel, Recht, Parrilo, Candès, ...

8 / 17

slide-9
SLIDE 9

minimize

Q, X

kQk∗ subject to A X + X A∗ + Q = 0 trace ( Tk X ) = gk, k = 1, . . . , N X ⌫ 0 velocity correlation matrix

9 / 17

slide-10
SLIDE 10

minimize

Q, X

kQk∗ subject to A X + X A∗ + Q = 0 trace ( Tk X ) = gk, k = 1, . . . , N X ⌫ 0

I Customized algorithms Zare, Jovanovi´ c, Georgiou, ACC ’14 Zare, Jovanovi´ c, Georgiou, ACC ’15, Submitted

10 / 17

slide-11
SLIDE 11

Turbulent channel flow

Φ(κ) := lim

t → ∞ E (v v∗)

v = [ u v w ]T

I Covariance completion

+

11 / 17

slide-12
SLIDE 12

One-point velocity correlations

y y

I Direct Numerical Simulations I Covariance completion problem (CCP)

  • 12 / 17
slide-13
SLIDE 13

Two-point velocity correlations

Simulations CCP

Φ11 Φ33

13 / 17

slide-14
SLIDE 14

Low rank solution

Q = (A X + X A∗)

i σi (Q) feasibility problem − → no clear cut σi (Q) covariance completion problem − → low-rank solution

14 / 17

slide-15
SLIDE 15

Filter design

key result: filter design using ( linearized dynamics completed correlations

15 / 17

slide-16
SLIDE 16

Verification in linear stochastic simulations

I Rτ = 180; kx = 2.5, kz = 7

uu y uv y Direct Numerical Simulations Linear Stochastic Simulations

  • Ek

t

16 / 17

slide-17
SLIDE 17

Concluding remarks

I Acknowledgments

NSF Award CMMI 1363266 2014 CTR Summer Program Minnesota Supercomputing Institute

17 / 17