Passive control of fluid-structure instabilities by means of - PowerPoint PPT Presentation

Passive control of fluid-structure instabilities by means of piezo-shunts Marco CARINI, Jean-Lou PFISTER & Olivier MARQUET AEROFLEX project 16th European Turbulence Conference - KTH Stockholm

Piezo-electricity : a wide range of applications Piezo-electricity : Electric response of certain mate- rials to a mechanical stress due to their microscopic structure Sensing & Actuation from everyday life . . . . . . to the Aerospace Research Focus on : using piezos to control fluid-structure global instabilities ETC16 - Carini, Pfister & Marquet 2 / 18 23rd August 2017

2D Model problem Fluid region Solid region Rigid cylinder + elastic plate + piezo patches. Incompressible flow at Re = U ∞ D/ν = 80 . Solid-to-fluid density ratio ρ = 50 and plate bending stiffness K = 0 . 3 . Electromechanical coupling coefficient k e = 0 . 57009 . ETC16 - Carini, Pfister & Marquet 3 / 18 23rd August 2017

Modelling framework Incompressible Navier-Stokes equations on the deforming domain : ( u T )) ∂ u u ⊗ u + p I − 1 ( ∂t + ∇· u + ∇ = 0 , ∇ Re ∇· u = 0 , Linear isotropic elastic material under small deformations : ρ∂ 2 ξ s ∂t 2 − ∇· σ s ( ξ s ) = 0 , Piezo-patch modelling : ρ∂ 2 ξ s ∂t 2 − ∇· σ s ( ξ s , V e ) = 0 , ∇· d ( V e , ξ s ) = 0 . Arbitrary Lagrangian Eulerian ( ALE ) formulation. Monolithic approach using FEM discretisation. ETC16 - Carini, Pfister & Marquet 4 / 18 23rd August 2017

Electro-mechanical coupling Discretized piezo-structure eqs. M d 2 ξ s dt 2 + K ξ s − K p ∆ V e = 0 , K T p ξ s + C p ∆ V e = Q e , C p is the equivalent piezo-patches capacitance. K p is the electromechanical coupling matrix bewteen σ xx and the transverse s electric field e y . Parallel connection ∆ V t e = ∆ V b e = ∆ V e . Opposite poling y -direction ⇒ only bending modes are affected : K T p ξ s = 0 , for pure traction/compression ETC16 - Carini, Pfister & Marquet 5 / 18 23rd August 2017

Connecting a R-shunt circuit 1st order electr. dynamics dQ e + 1 τ e Q e = 0 , dt where τ e = R e C p is the characteristic electric time . Short-circuit ⇒ No electro-mechanical coupling ⇒ pure fluid-structure behaviour . Open-circuit ⇒ Maximal electro-mechanical coupling. ETC16 - Carini, Pfister & Marquet 6 / 18 23rd August 2017

Fluid-Structure-Electric System ∂ q f ∂t − R f ( q f , q s ) = 0 , on Ω r B f Fluid Eqs. f B s ∂ q s on Ω r ∂t − R s ( q s , q s ) = 0 , Solid Eqs. s B p d q e dt − R p ( q e , q s ) = 0 , Electr. Eqs. where ( u ) ( ) ( Q e ) ξ s q f = , q s = , q e = . p ∂ ξ s /∂t V e and in compact abstract from : B ∂ q ∂t = R ( q ) . where q = { q f , q s , q e } T . ETC16 - Carini, Pfister & Marquet 7 / 18 23rd August 2017

Global stability analysis Nonlinear fixed equilibrium solution q B : R ( q B ) = 0 . Linearized perturbations around q B : q ( x ) e λt , q ( x , t ) = q B ( x ) + ϵ ˆ with ϵ ≪ 1 . Generalised Eigenvalue problem : λ B ˆ q = A ˆ q , where :  A ff I fs 0  � A = ∂ R ( q ) �   = I fs A ss A sp  . �   ∂ q �  q B 0 A ps A pp ETC16 - Carini, Pfister & Marquet 8 / 18 23rd August 2017

Global stability results 1.2 Base flow 0.9 ω 0.6 Global spectrum for Short and Open circuit case. Investigate the mode transition : 0.3 ▶ based on the added electrical stiffness effect. ▶ by varying τ e . 0 -0.2 -0.1 0 0.1 σ ETC16 - Carini, Pfister & Marquet 9 / 18 23rd August 2017

Leading fluid-structure-electric modes Short-circuit Open-circuit When using the same normalization, the maximum y -displacement is increased by 3 orders of magnitude in the open-circuit case. ETC16 - Carini, Pfister & Marquet 10 / 18 23rd August 2017

Structure mode kinematic analysis Projection of q s on the structure free-vibration modes ω 1 = 0 . 84 ω 2 = 5 . 25 ω 3 = 14 . 59 Projection coefficient amplitude Mode 1 Mode 2 Mode 3 Mode 4 2 . 89 × 10 − 3 1 . 99 × 10 − 4 5 . 29 × 10 − 5 Short-circuit mode 1 . 00 1 . 53 × 10 − 1 2 . 61 × 10 − 2 1 . 14 × 10 − 2 Open-circuit mode 0 . 99 ETC16 - Carini, Pfister & Marquet 11 / 18 23rd August 2017

Structure mode kinematic analysis (continued) Projection of q s on the piezo-structure free-vibration modes ω 1 = 1 . 33 ω 2 = 7 . 64 ω 3 = 17 . 66 Projection coefficient amplitude Mode 1 Mode 2 Mode 3 Mode 4 1 . 70 × 10 − 3 4 . 42 × 10 − 4 3 . 39 × 10 − 5 Open-circuit mode 1 . 00 ETC16 - Carini, Pfister & Marquet 12 / 18 23rd August 2017

Structure mode dynamic analysis For the free-vibrating piezo-elastic plate in open-circuit we have : − ω 2 M ˆ ξ s + ( K + K p C − 1 p K T p ) ˆ ξ s = 0 By projection onto the base of free-vibrating modes of the elastic plate only, X s , we obtain ω 2   0 , 1 ω 2  0 , 2  ω 2 ˆ ˆ ζ + ∆Ω p ˆ with ∆Ω p = X H s K p C − 1 p K T ζ = ζ , p X s  ...      ω 2 0 ,n Truncation at the first 6 modes yields ω 1 = 1 . 4 (compared to ω 1 , exact = 1 . 33 ) and abs ( ˆ 0 . 9834 , 0 . 1782 , 0 . 0312 , 0 . 0134 , 3 . 33 × 10 − 10 , 0 . 0050 ( ) ζ 1 ) = ⇒ the first free-vibrating piezo mode essentially results from the combination of the first two free vibration modes of the elastic plate. ETC16 - Carini, Pfister & Marquet 13 / 18 23rd August 2017

Varying the electrical resistance Leading eigenvalue 0.06 1.1 0.03 1 σ ω 0 0.9 -0.03 0.8 -0.06 0.7 -6 -4 -2 1 2 3 -6 -4 -2 1 2 3 log 10 τ e log 10 τ e ETC16 - Carini, Pfister & Marquet 14 / 18 23rd August 2017

Varying the electrical resistance (continued) Increasing τ e 1.2 1 ω 0.8 0.6 -0.2 -0.15 -0.1 -0.05 0 0.05 σ ETC16 - Carini, Pfister & Marquet 15 / 18 23rd August 2017

System energy analysis Leading mode 12 × 10 -4 1.2 0.8 8 E s % P e % 0.4 4 0 0 -6 -4 -2 1 2 3 -6 -4 -2 1 2 3 log 10 τ e log 10 τ e The white area denotes the stabilization range. The red dashed line corresponds to the maximal damping of the leading mode. ETC16 - Carini, Pfister & Marquet 16 / 18 23rd August 2017

Conclusions Structure-added electrical stiffness effect as expected. Fluid-structure mode ( mainly driven by the unstable fluid dynamics ) controlled by exploiting the piezo electro-mechanical coupling . Open/close circuit mode selection explained by continuosly varying R e : ▶ the original fluid-structure mode is increasingly damped as R e is increased. ▶ for large enough values of R e a second mode is destabilized ( water-bed effect ). ▶ effective stabilization by passive control within a finite range of R e values. Ongoing developments Varying elastic parameters to address other types of fluid-structure modes. Introducing a second-order electric dynamics. Active feedback control. ETC16 - Carini, Pfister & Marquet 17 / 18 23rd August 2017

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