Identifying the frequency selection of fluid/structure instabilities - PowerPoint PPT Presentation

Identifying the frequency selection of fluid/structure instabilities when the interaction is large Olivier Marquet 1 & Lutz Lesshafft 2 1 Department of Fundamental and Experimental Aerodynamics 2 Laboratoire d’Hydrodynamique, CNRS-Ecole Polytechnique 11 th Euromech Fluid Mechanics Conference 12-16 September 2016, Sevillel, Spain

Context Flow-induced structural vibrations Aeronautics Offshore-marine industry Civil engineering Predict the onset of vibrations (based on stability analysis) and control them The origin of fluid/structure instabilities ? 2

A model problem spring-mounted cylinder flow One spring in the cross-stream direction Reynolds number Density ratio Structural frequency Structural damping �� = � � � � = � � � � � � � � = 40 � ⁄ � � = � 3

Self-sustained oscillations �� = 40 , � = 10 , � � = 0 No oscillation � � = 0.6 Weak oscillation � � = 0.7 � � = 0.9 Strong oscillation No oscillation � � = 1.1 4

Outlines 1 – Stability analysis of the fluid/structure problem a – Operator definition and formalism b – Results for weak and strong interaction 2 – Identification the driving dynamics of coupled modes a – Operator decomposition approach b – Results for strong interaction 5 Titre présentation

Stability analysis of the coupled fluid/solid problem Stability of the steady solution (fixed cylinder) ′((, )) = ( � , � ) ( � *+, - . + /. /. Growth rate/frequency Fluid/solid components Linearized fluid equations � � (��) � � � = (! + #$) % 0 � � � � � �� � � � � (� � , � � ) 0 & Damped harmonic oscillator Cossu & Morino (JFS, 2000) 6

Results – Eigenvalue spectrum � = 10 1 � � = 0.75 Eigenvalue spectrum Are the coupled modes driven by the fluid or the solid dynamics ? 7

Results – Components of eigenvector Infinite mass ratio - weak interaction Fluid component 2 � = 3 � ≠ 0 2 � = 3 � = 0 Solid component Fluid (wake) mode Solid mode For small mass ratio - strong interaction ? 2 � = 3 � ≠ 0 8

Results – Variation of stiffness Infinite mass ratio – weak interaction � � = 0.75 0.4 < � � < 1.2 Solid mode : $ ∼ � � (= structural frequency) Wake modes : $ ∼ $ 8 (= vortex-shedding frequency) 9

Results – Variation of stiffness Finite mass ratio – strong interaction � = 10 � = 200 Destabilization of solid branch Destabilization of fluid branch? The two branches exchange their « nature » for small mass ratio Zhang et al (JFM 2015), Meliga & Chomaz (JFM 2011) 10

Outlines 1 – Stability analysis of the fluid/structure problem a – Operator definition and formalism b – Results (various mass ratio / structural frequency) 2 – Identification of the driving dynamics a – Operator decomposition approach b – Results 11

The eigenvalue problem Eigenvalue problem - Coupled operator � � (��) � � � = (! + #$) % 0 � � � � � �� � � � � (� � , � � ) 0 & Infinite mass ratio � � (��) � � � = (! + #$) % 0 � � � � 0 � � (� � , � � ) 0 & Fluid mode = eigenvalue/vector of � � Solid mode = eigenvalue/vector of � � 12

From operator to eigenvalue decomposition Operator decomposition � = � 9 + � : = ; In general, is not an eigenmode of � 9 or � : , so � : = ; : + < � 9 = ; 9 + < : 9 < 9 ≠ 0, < : ≠ 0 but < 9 = −< : = < with residuals Eigenvalue decomposition ; 9 + ; : = ; How to compute the eigenvalue contributions ; 9 /; : ? 13

Computing eigenvalue contributions Expansion of the residual on the set of other eigenmodes ? < = A < ? ? � 9 = ; 9 + A < ? ? ? ? Orthogonal projection on the mode using the adjoint mode + +@ (� 9 ) = ; 9 ( +@ ) + A < ? ( +@ ? ) ? = 0 = 1 Bi-orthogonality Normalisation Adjoint mode-based decomposition ; = ; 9 + ; : ; 9 = +@ (� 9 ) ; : = +@ (� : ) 14

Why this particular eigenvalue decomposition? For an identical decomposition of the operator, other eigenvalue decompositions are possible Non-orthogonal projection on the mode Direct mode-based decomposition ; = ; B 9 + ; B : B 9 = @ (� 9 ) B : = @ (� : ) ; ; But it includes contributions from other eigenmodes � 9 = ; 9 + A < ? ? � : = ; : − A < ? ? ? ? ? ( @ ? ) ; B 9/: = ; 9/: ± A < ≠ 0 ? M.Juniper (private communication) 15

Application to the spring-mounted cylinder flow � � � � � = ; % 0 � � � � � �� � � � � 0 & Adjoint mode-based decomposition ; = ; � + ; � Fluid contribution Solid contribution +@ (� � � + � +@ (� � � + � �� � � � ) ; � = � � � ) ; � = � Adjoint solid component Adjoint fluid component 16

Application to the spring-mounted cylinder flow � � � � � = ; % 0 � � � � � �� � � � � 0 & Adjoint mode-based decomposition ; = ; � + ; � Fluid contribution Solid contribution +@ � ) +@ � ) ; � = ;( � ; � = ; ( � Direct and adjoint Direct and adjoint fluid components solid components 17

Stability results for D = EFF – Solid branch Frequency Growth rate 18

Solid branch: Frequency decomposition $ = $ � + $ � Solid contribution Fluid contribution $ � = ℑ(; � ) $ � = ℑ(; � ) Solid Fluid The frequency is selected by the solid dynamics 19

Solid branch: Growth rate decomposition ! = ! � + ! � Solid contribution Fluid contribution ! � = ℜ(; � ) ! � = ℜ(; � ) Large and opposite contributions in the unstable region 20

Solid branch: Growth rate decomposition ! = ! � + ! � Solid contribution Fluid contribution ! � = ℜ(; � ) ! � = ℜ(; � ) ! > 0 Low stiffness High stiffness | ! � | > |! � | |! � | > |! � | destabilization by the destabilization by solid contribution the fluid contribution 21

Local (fluid) contribution to the growth Stabilizing region Destabilizing region + � � Phase difference between and Schmid & Brandt (AMR 2014) 22

Stability results for D = KF Eigenvalue spectrum WM SM Frequency of the Wake Mode branch 23

« So called » fluid branch: Frequency decomposition $ = $ � + $ � Solid contribution Fluid contribution $ � = ℑ(; � ) $ � = ℑ(; � ) High stiffness Low stiffness $ ∼ $ � $ � ∼ $ � $ ∼ $ � Frequency selection Frequency selection by the solid dynamics by the fluid dynamics 24

Conclusion Operator decomposition approach applied to coupled fluid/solid • modes No need to vary the parameters (mass ratio or stiffness), need to • determine the adjoint modes. Results similar to « wavemaker » analysis (structural sensitivity) • Not a variation of eigenvalues but a decomposition of eigenvalues • Extension to more complex solid dynamics (Jean-Lou Pfister - PhD) • Thank you 25

Pure modes (infinite mass ratio) Direct Adjoint L @ ; � � ; ∗ O � O � 0 = L � �� = @ � � O � M @ O � � 0 M �� P = L @ O � P ∗ O � ; ; � = L � � Fluid modes P = � ∗ & − M @ O � P � = 0 @ O � ; � �� P = M @ O � ; � = M � P ∗ O � ; � Solid modes P = 0 (;& − L) � = � �� � O � 26 Titre présentation

Projection of coupled problem on pure fluid modes (; & − L) � = � �� � (;& − M) � = � �� � �� � P@ ; & − L � + O � P@ ; & − M � − O � P@ � P@ � �� � �� � = � �� O � O � P @ P @ P − L @ O � P − M @ O � P − � P@ � �� � ; ∗ O � � + ; ∗ O � @ O � � = � �� O � �� P@ � ) + ; − ; P@ � ) = � �� O � P@ � �� � ; − ; � (O � � (O � P@ � �� � � �� O � ; − ; � = P@ � + O � P@ � ) (O � 27 Titre présentation

Projection of coupled problem on pure solid modes (; & − L) � = � �� � (;& − M) � = � �� � �� � P@ ; & − L � + O � P@ ; & − M � − O � P@ � P@ � �� � �� � = � �� O � O � P @ P @ P − L @ O � P − M @ O � P − � P@ � �� � ; ∗ O � � + ; ∗ O � @ O � � = � �� O � �� P@ � ) = � �� O � P@ � �� � (; − ; � )(O � P@ � �� � ; − ; � = � �� O � P@ � O � 28 Titre présentation

Direct-based decomposition of the unstable mode Frequency � = 200 Fluid Solid Growth rate Solid Fluid 29 Titre présentation

Infinite mass ratio - Fluid Modes � � � � � = ; % 0 � �� = 0 �� � � � F � � 0 & Adjoint mode-based decomposition ; = ; � + ; � Fluid contribution Solid contribution +@ (� � � + � +@ (� � � + � �� � �� � ) ; � = � �� � ) ; � = � � �� = 0 Q R = F Fluid Modes +@ � � � = ; ; � = 0 ; � = � OK 30