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Prediction of flutter instability in turbulent flow based on Linear Stability Analysis J.Moulin, O.Marquet, D.Sipp Office National dEtudes et de Recherches Arospatiales Dpartement dArodynamique, Arolasticit et Aroacoustique

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SLIDE 1

Prediction of flutter instability in turbulent flow based on Linear Stability Analysis

J.Moulin, O.Marquet, D.Sipp

Office National d’Etudes et de Recherches Aérospatiales Département d’Aérodynamique, Aéroélasticité et Aéroacoustique

Funded by ERC Starting Grant BIFD, Houston,12 July 2017

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SLIDE 2

Introduction

2

Fluid-Structure system exhibit various types of instability phenomenons

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SLIDE 3

Introduction

3

Fluid-Structure system exhibit various types of instability phenomenons Accurate fluid-structure modeling is needed

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SLIDE 4

Introduction

4

Streamlined body - Attached flow Heaving and pitching involved No robust modelling Experiments / Time-marching simulations Turbulent flows Potential flow modelling Analytical methods [Theodorsen 1935] Bluff body - Separated flow Only pitching motion involved in [Païdoussis 2011]

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SLIDE 5

Introduction

5

Turbulent flows with a turbulent flow modelled with Reynolds Averaged Navier Stokes (RANS) equations Linear stability analysis of the coupled fluid-structure interaction problem Proposed method Streamlined body - Attached flow Heaving and pitching involved No robust modelling Experiments / Time-marching simulations Potential flow modelling Analytical methods [Theodorsen 1935] Bluff body - Separated flow Only pitching motion involved

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SLIDE 6

6

Part 1 – Configuration and Modelling Part 2 –An elongated plate mounted on two springs Part 3 –A short plate mounted on two springs

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SLIDE 7

Part 1 : Configuration and Modelling

7

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SLIDE 8

Aeroelastic configuration

8

Heaving and pitching rigid body in turbulent incompressible flows , ,

Elastic axis

  • ℎ

+

Torsional spring Mass center Translational spring

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SLIDE 9

Aeroelastic configuration

9

Two aspect ratios are investigated

=

  • 5

23

« Bridge type » « Airfoil type »

  • Short plate

Elongated plate =

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SLIDE 10

10 10

Modelisation

Non-dimensional numbers :

=

  • =
  • ∗ =

1

  • Shape parameters :

=

  • Fluid parameter :

Solid parameter : Coupling parameter :

  • =
  • ! =
  • "#$ : non-dimensional

heaving natural frequency

"#% : non-dimensional

pitching natural frequency

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SLIDE 11

11 11

Modelisation

Non-dimensional numbers :

= 0,08 =

  • ∗ =

1

  • Shape parameters :

= 2,7 ⋅ 10+

Fluid parameter : Solid parameter : Coupling parameter :

  • = 10+

! = 0,8

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SLIDE 12

12 12

Fluid-structure modelling

,-. ,/ = 0. -. + 1.2(-., -2)

Coupled fluid-structure equations Fluid variables Solid variables

  • .
  • 2

5-2 5/ = 12. -. + 02 -2

Fluid equation Solid equation

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13 13

Fluid-structure modelling

,-. ,/ = 0. -. + 1.2(-., -2)

Fluid dynamics

RANS approach Spalart-Allmaras turbulent model

  • . = 6, 7, 8, ̃ :

5-2 5/ = 12. -. + 02 -2

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SLIDE 14

14 14

Fluid-structure modelling

,-. ,/ = 0. -. + 1.2(-., -2)

Fluid dynamics

RANS approach Spalart-Allmaras turbulent model

Solid dynamics

2-DOF rigid body ℎ; +

< ℎ + ; = 0

; +

< + =() ℎ; = 0

  • 2 = ℎ, , ℎ>, > :
  • . = 6, 7, 8, ̃ :

5-2 5/ = 12. -. + 02 -2

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SLIDE 15

15 15

Fluid-structure modelling

,-. ,/ = 0. -. + 1.2(-., -2)

Fluid dynamics

RANS approach Spalart-Allmaras turbulent model

Solid dynamics

2-DOF rigid body

Fluid-to-solid coupling

Fluid force - Lift Fluid moment

Solid-to-fluid coupling

Interface conditions Non-inertial volumic terms [Mougin et al. 2002]

  • . = 6, 7, 8, ̃ :
  • 2 = ℎ, , ℎ>, > :

5-2 5/ = 12. -. + 02 -2

ℎ; +

< ℎ + ; = 0

; +

< + =() ℎ; = 0

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SLIDE 16

16 16

Linear stability analysis

Perturbation decomposition

  • . ?, / = @. ? + A -.

B ?, /

  • 2 / = # + A -2

B /

Fluid Solid

Modal decomposition

  • .

B ?, C = -

  • D. ? EFG + H. H
  • 2

B C = -

D2 EFG + H. H

Fluid Solid

Two classical ingredients :

base state perturbation

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SLIDE 17

17 17

Linear stability analysis

Coupled eigenvalue problem

J = ℜ[M]

Growth rate Frequency

= ℑ[M]

Fluid-solid mode

(- D., - D2)

M P - D.

  • D2

=

  • D.
  • D2

Fluid-structure Jacobian matrix Mass matrix (spatial discretizarion)

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Part 2 : An elongated plate mounted on two springs (AR=23)

18

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19 19

Steady base flow

Axial velocity ( = 27500) Turbulent to kinematic viscosity ratio

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20 20

Steady base flow

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SLIDE 21

21 21

Steady base flow

Lift coefficient Moment coefficient ,QR ,S T

UV

,QW ,S T

UV

Present study 7,0 1,8 Potential flows 7,0 1,7

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SLIDE 22

22 22

Steady base flow

Lift coefficient Moment coefficient Small influence of detached areas on slopes at S = 0° ,QR ,S T

UV

,QW ,S T

UV

Present study 7,0 1,8 Potential flows 7,0 1,7

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SLIDE 23

23 23

Linear Stability Analysis

Coupled fluid-structure system

MY- D = Z- D Z = Z.. Z.2 Z2. Z22

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SLIDE 24

24 24

Linear Stability Analysis

Uncoupled fluid system

M

Y..-

  • D. = Z-

D.

Fluid system spectrum

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SLIDE 25

25 25

Linear Stability Analysis

Uncoupled fluid system

M

Y..-

  • D. = Z-

D.

Fluid system spectrum

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SLIDE 26

26 26

Linear Stability Analysis

Uncoupled fluid system

M

Y..-

  • D. = Z-

D. = 20,7 ([/\ ≃ 0,14)

Fluid system spectrum

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SLIDE 27

27 27

Linear Stability Analysis

Uncoupled fluid system

M

Y..-

  • D. = Z-

D. = 20,7 ([/\ ≃ 0,14)

Fluid system spectrum

Uncoupled Fluid spectrum

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SLIDE 28

28 28

Linear Stability Analysis

Uncoupled solid system

MY22- D2 = Z- D2 ∗ = 5

Inertial coupling bewteen heaving and pitching

  • = 0,76
  • = 1,08

Uncoupled Fluid spectrum Uncoupled Solid spectrum Low-frequency mode High-frequency mode

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SLIDE 29

29 29

Linear Stability Analysis

Uncoupled fluid and solid systems

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SLIDE 30

30 30

Linear Stability Analysis

Coupled fluid-structure system

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SLIDE 31

31 31

Linear Stability Analysis

Coupled fluid-structure system

MY- D = Z- D Z = Z.. Z.2 Z2. Z22

∗ = 5

  • = 10+
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SLIDE 32

32 32

Linear Stability Analysis

Coupled fluid-structure system

M- D = Z- D Z = Z.. Z.2 Z2. Z22

Mode kinetic energy : `= 1 `

= 10a

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SLIDE 33

33 33

Linear Stability Analysis

Coupled fluid-structure system

M- D = Z- D Z = Z.. Z.2 Z2. Z22

Mode kinetic energy : `= 1 `

= 4,1

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SLIDE 34

34 34

Linear Stability Analysis

Coupled fluid-structure system

M- D = Z- D Z = Z.. Z.2 Z2. Z22

Mode kinetic energy : `= 1 `

= 1,1

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SLIDE 35

Aeroelastic instabilities of a 2-DOF elongated plate

35

∗ → 0 : uncoupled case

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SLIDE 36

Aeroelastic instabilities of a 2-DOF elongated plate

36

Coupled mode flutter

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Aeroelastic instabilities of a 2-DOF elongated plate

37

Coupled mode flutter

∗ = 5

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SLIDE 38

Aeroelastic instabilities of a 2-DOF elongated plate

38

Coupled mode flutter

∗ = 5

VIV (not investigated here) cdc

∗

≃ 1 2e [/\ ≃ 0,05

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SLIDE 39

Validation

Comparison to Theodorsen model

39

fghG

∗

  • Present study

2,9 0,81 Theodorsen 3,0 0,81

  • Modelisation validated against

classical Theodorsen flutter theory

  • Maginal role of detached areas on

instability thresholds

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SLIDE 40

Part 3 : An short plate mounted on two springs (AR=5)

40

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SLIDE 41

41

Steady Base Flow

Large leading-edge detached areas

Non-negligible effect on steady aerodynamic coefficient

0,75

,QR ,S T

UV

,QW ,S T

UV

Present study 9,15 0,95 Potential flows 8,4 1,7

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SLIDE 42

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Aeroelastic instabilities of a 2-DOF short plate

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SLIDE 43

43

VIV (not investigated here) cdc

∗

≃ 1 2e [/\ ≃ 0,2

Aeroelastic instabilities of a 2-DOF short plate

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SLIDE 44

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VIV (not investigated here) cdc

∗

≃ 1 2e [/\ ≃ 0,2

Aeroelastic instabilities of a 2-DOF short plate

Single mode high frequency flutter

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Aeroelastic instabilities of a 2-DOF short plate

Single mode high frequency flutter

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Aeroelastic instabilities of a 2-DOF short plate

Single mode high frequency flutter Single mode low frequency flutter

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Aeroelastic instabilities of a 2-DOF short plate

Single mode high frequency flutter Single mode low frequency flutter

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SLIDE 48

Aeroelastic instabilities of a 2-DOF short plate

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Low frequency flutter mode

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Aeroelastic instabilities of a 2-DOF short plate

Single mode high frequency flutter Single mode low frequency flutter

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Aeroelastic instabilities of a 2-DOF short plate

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Low frequency flutter mode High frequency flutter mode

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SLIDE 51

Comparison to Theodorsen …

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ijklC

∗

"/"#%

Present study (low frequency branch) 0,98 0,77 Present study (high frequency branch) ~1,4 1,08 Theodorsen 4,3 1,03

  • Clear difference bewteen RANS and Theodorsen
  • Well known phenomena : single-mode flutter is difficult to model
  • Neither Theodorsen nor other simple method work

Interest of using a full RANS modelling

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Conclusions :

  • Single-mode and coupled-mode flutter in turbulent flow have been

investigated through global linear stability analysis …

  • Streamlined body : coupled-mode flutter
  • Bluff body : single-mode flutter
  • A full RANS modelling of the fluid has been used
  • The role of recirculation areas in stability has been discussed
  • Theodorsen’s model can be extended to the case of limited detached

areas

Conclusion & Perspectives

52

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Conclusion & Perspectives

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Perspectives :

  • Experimental results on the case = 5 for validation
  • Consider flexible structures
  • 3D configurations
  • Towards stabilization strategies of those unstable modes