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Fluid-solid-electric stability analysis for the control of a flexible splitter plate M. Carini, J.-L. Pfister and O. Marquet ONERA-DAAA, Meudon XXIII Conference The Italian Association of Theoretical and Applied Mechanics, 4 th -7 th

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SLIDE 1

Fluid-solid-electric stability analysis for the control of a flexible splitter plate

  • M. Carini, J.-L. Pfister and O. Marquet

ONERA-DAAA, Meudon

XXIII Conference – The Italian Association of Theoretical and Applied Mechanics, 4th-7th September, Salerno, Italy

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SLIDE 2

Introduction

Temporal simulation of fluid-solid interaction

  • 1. Investigate the dynamics based on fluid-solid stability analysis
  • 2. Control by stabilisation with piezo-electric patches
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SLIDE 3

Outlines

  • 1. Configuration, physical and numerical models
  • 2. Fluid-solid stability analysis
  • 3. Stabilization using piezoelectric patches

3.1 Fluid-solid-electric stability analysis 3.2 Results for short- and open-circuit configurations 3.3 Results for resistive

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SLIDE 4

Flow configuration and nondimensional parameters

Rigid cylinder Elastic plate Uniform incoming flow

  • Incompressible laminar flow - Reynolds number
  • Elastic plate
  • Bending stiffness
  • Poisson coefficient
  • Solid-to-fluid density ratio
  • 80

0.3 / 50 0.4

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SLIDE 5

Physical modelling

  • Fluid model – Eulerien :

Incompressible Navier-Stokes equations

  • Solid model - Lagrangian:

Linear isotropic material under small deformations

  • Fluid-solid interface

Stress and velocity continuity

Ω

Time-dependent domain Time-independent (reference) domain - Ω

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SLIDE 6

Fluid/structure coupling and numerical treatment

Arbitrary Lagrangian Eulerian mapping

Time-dependent domains Time-independent reference domains

Solid domain The displacement variable maps

Ω

  • Ω()

Reference domain Deformed domain

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SLIDE 7

Fluid/structure coupling and numerical treatment

Arbitrary Lagrangian Eulerian mapping

Time-dependent domains Time-independent reference domains

Fluid domain An artificial displacement variable is introduced

Ω

  • Ω()

Reference domain Deformed domain

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SLIDE 8

Fluid/structure coupling and numerical treatment

Arbitrary Lagrangian Eulerian mapping

Time-dependent domains Time-independent reference domains

Fluid domain An artificial displacement variable is introduced which satisfies an arbitrary extension equation

() 0 () ()

in Ω

  • n Γ
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SLIDE 9

Fluid/structure coupling and numerical treatment

Arbitrary Lagrangian Eulerian mapping

Time-dependent domains Time-independent reference domains

New variable , the fluid mesh displacement Additional equation to spread the displacement at the interface

() 0 () () Ω

  • Γ
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SLIDE 10

Outlines

  • 1. Configuration, physical and numerical models
  • 2. Fluid-solid stability analysis
  • 3. Stabilization using piezoelectric patches

3.1 Fluid-solid-electric stability analysis 3.2 Results for short- and open-circuit configurations 3.3 Results for resistive

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SLIDE 11

Steady solutions of the fluid/solid equations

The nonlinear coupled fluid/structure problem is

!" ! (")

written in a time-independent reference domain Ω Ω

∪ Ω

  • "($, ) &, ', , ,

(

)

"* 0 Steady solutions

Fluid velocity Solid displacement Small compression Flow recirculation

0.015 80

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SLIDE 12

Lineary stability of the fluid/solid equations

" ,, "* $ + . " / , (012 3)4 + " /∗ , (062 3)4 7 + 8 9 : " / + ; "* " / < Linear stability analysis

7: growth rate 9: frequency

Eigenvalue problem

; "* " / ; ; ; ; " / " /

" / (& /, '̂, >) " / ( >, ( >

)

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SLIDE 13

Results – Eigenvalues and eigenmodes

80 0.015

Unsteady mode Steady mode Two unstable modes Eigenvalue spectrum

9 7 7 + 8 9 : " / + ; "* " / <

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SLIDE 14

Outlines

  • 1. Configuration, physical and numerical models
  • 2. Fluid-solid stability analysis
  • 3. Stabilization using piezoelectric patches

3.1 Fluid-solid-electric stability analysis 3.2 Results for short- and open-circuit configurations 3.3 Results for resistive

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SLIDE 15

Piezoelectric patches

  • Two piezoelectric patches
  • Continuous modelling of one piezo-patch

Rigid cylinder Elastic plate

!? !? + @ ⋅ B , C

D 0

@ ⋅ E C

D,

The Cauchy stress tensor is modified to take into account the electro-mechanical coupling effects

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SLIDE 16

Piezoelectric patches

Discrete modelling of two piezo-patches (Thomas et al. 2009) which are

  • connected in parallel ΔC

D4 ΔC DG C D

  • polarized in the transverse direction H 0
  • with opposive direction I

4 JI G

Rigid cylinder Elastic plate

K !? !? + J L C

D 0

L

) + ML C D ND

ML: equivalent piezo-patches capacitance L: electro-mechanical coupling matrix

(here only between 7

HHand H)

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SLIDE 17

Piezo-shunt configuration – Resistive circuit

QND Q + 1 RD ND 0

Characteristic electric time RD DML

Short-circuit case Open-circuit case D 0 C

D 0

D → ∞ ND 0

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SLIDE 18

Short-circuit configuration K !? !? + J L C

D 0

C

D 0

K !? !? + 0

No electro-mechanical coupling

Short-circuit configuration = Fluid-solid configuration

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SLIDE 19

Open-circuit configurations C

D

ND 0 L

) + ML C D ND

C

D J ML 6UL )

K !? !? + ( + LML

6UL )) 0

Piezo-patches have an added-stiffness effect

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SLIDE 20

Fluid-solid-electric stability analysis

Fluid-solid-electric eigenvalue problem

; "* " / ; ; < ; ; ;V < ;V ;VV " / " / " /V

" / (& /, '̂, >) " / ( >, ( >

)

" /V (C >

D, N

>D) 7 + 8 9 : " / + ; "* " / <

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SLIDE 21

Results – Eigenvalue and eigenmodes

80 0.3 50 WD 0.57

Eigenvalue spectrum Unstables eigenmodes Fluid-solid (short-circuit) Fluid-solid-electric (open-circuit) Maximal displacement increased by 3 orders of magnitude !

9 0.85 ? 9 1.33 ?

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SLIDE 22

Free-vibration modes

Free-vibration modes of the elastic plate Z[ <. \] 9? 5.25 9_ 14.59 Free-vibration modes of the elastic plate with piezo-patches Z[ [. aa 9? 7.64 9_ 17.66

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SLIDE 23

Free-vibration modes – Frequency comparison

Free-vibration modes of the elastic plate Z[ <. \] 9? 5.25 9_ 14.59 Free-vibration modes of the elastic plate with piezo-patches Z[ [. aa 9? 7.64 9_ 17.66 Z <. \] Z [. aa

Fluid-solid (short-circuit) Fluid-solid-electric (open-circuit)

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SLIDE 24

Free-vibration modes – Amplitude comparison

Free-vibration modes of the elastic plate c[ <. dded f? 0.0029 f_ 0.0002 Free-vibration modes of the elastic plate with piezo-patches c[ <. ddgd f? 0.0017 f_ 0.0004

Fluid-solid (short-circuit) Fluid-solid-electric (open-circuit)

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SLIDE 25

Results for the piezo R-shunt configuration

Behaviour of the leading eigenvalue when varying the resistance

Growth rate Frequency

Stabilization in a range

  • f electric resistance

Jump of the frequency

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SLIDE 26

Results - Piezo-shunt configuration

Behaviour of the leading eigenvalue when varying the resistance Small hV : stabilization of the fluid-solid eigenmode Large hV : destabilization of the fluid-solid-electric eigenmode

Eigenvalue spectrum Short-circuit D 0 Open-circuit D → ∞

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SLIDE 27

Perspectives

  • Passive control

Investigate the effect of piezo-patches in other configurations

  • Case with the nstable steady mode
  • Case where the piezo-patches have other

material properties

  • Introduce a second-order electric dynamics (add inductance

to the resistive circuit)

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