Fluid-solid-electric stability analysis for the control of a - PowerPoint PPT Presentation

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 September, Salerno, Italy

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

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

Flow configuration and nondimensional parameters Rigid cylinder Elastic plate Uniform incoming flow �� � � � � � 80 Incompressible laminar flow - Reynolds number • � Elastic plate • � � � 0.3 - Bending stiffness � � � 0.4 - Poisson coefficient � � /� � � 50 Solid-to-fluid density ratio •

Physical modelling Fluid model – Eulerien : • Incompressible Navier-Stokes equations Time-dependent domain Ω � � Solid model - Lagrangian: • Linear isotropic material under small deformations � Time-independent (reference) domain - Ω � • Fluid-solid interface Stress and velocity continuity

Fluid/structure coupling and numerical treatment A rbitrary L agrangian E ulerian mapping Time-dependent domains Time-independent reference domains Solid domain The displacement variable � � � maps Deformed domain Reference domain � Ω � (�) Ω �

Fluid/structure coupling and numerical treatment A rbitrary L agrangian E ulerian mapping Time-dependent domains Time-independent reference domains Fluid domain An artificial displacement variable � � � is introduced Deformed domain Reference domain � Ω � (�) Ω �

Fluid/structure coupling and numerical treatment A rbitrary L agrangian E ulerian mapping Time-dependent domains Time-independent reference domains Fluid domain An artificial displacement variable � � � is introduced which satisfies an arbitrary extension equation � � (�) � � � (�) � � � (�) � 0 � � in ٠� on à ��

Fluid/structure coupling and numerical treatment A rbitrary L agrangian E ulerian mapping Time-dependent domains Time-independent reference domains New variable � � � , the fluid mesh displacement Additional equation to spread the displacement at the interface � � (�) � � � (�) � � � (�) � 0 � à � ٠� ��

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

( Steady solutions of the fluid/solid equations The nonlinear coupled fluid/structure problem is !" ) !� � �(") "($, �) � &, ', � � , � � , � � � ∪ Ω � written in a time-independent reference domain Ω � � Ω � � Steady solutions � " * � 0 �� � 80 � � � 0.015 Fluid Flow velocity recirculation Small Solid compression displacement

Lineary stability of the fluid/solid equations Linear stability analysis / ∗ , � (062 3)4 / , � (012 3)4 + " " ,, � � " * $ + . " 7 : growth rate 9 : frequency Eigenvalue problem 7 + 8 9 : " / + ; " * " / � < ; �� ; �� ; �� " / � ; " * " / � ; �� / � " > > � ) > � , �( " / � � (& /, '̂, � " / � � (� � )

Results – Eigenvalues and eigenmodes 7 + 8 9 : " / + ; " * " / � < � � � 0.015 �� � 80 Eigenvalue spectrum 9 Unsteady mode 7 Two unstable modes Steady mode

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

Piezoelectric patches Rigid cylinder Elastic plate Two piezoelectric patches • Continuous modelling of one piezo-patch • � ! ? � � !� ? + @ ⋅ B � � � , C D � 0 @ ⋅ E C D , � � � 0 The Cauchy stress tensor is modified to take into account the electro-mechanical coupling effects

Piezoelectric patches Rigid cylinder Elastic plate Discrete modelling of two piezo-patches (Thomas et al. 2009) which are D4 � ΔC DG � C - connected in parallel ΔC D - polarized in the transverse direction � H � 0 4 � J� I G - with opposive direction � I K ! ? � � M L : e quivalent piezo-patches capacitance !� ? + � � � J � L C D � 0 � L : electro-mechanical coupling matrix ) � � + M L C � L D � N D HH and � H ) (here only between 7 �

Piezo-shunt configuration – Resistive circuit QN D Q� + 1 N D � 0 R D Characteristic electric time R D � � D M L Short-circuit case Open-circuit case � D � 0 C D � 0 � D → ∞ N D � 0

Short-circuit configuration C D � 0 K ! ? � � K ! ? � � !� ? + � � � J � L C D � 0 !� ? + � � � � 0 No electro-mechanical coupling Short-circuit configuration = Fluid-solid configuration

Open-circuit configurations C N D � 0 D ) � � + M L C � L D � N D 6U � L ) � � C D � J M L K ! ? � � 6U � L ) ) � � � 0 !� ? + ( � + � L M L Piezo-patches have an added-stiffness effect

Fluid-solid-electric stability analysis Fluid-solid-electric eigenvalue problem / + ; " * " / � < 7 + 8 9 : " ; �� ; �� < / � " ; �� ; �� ; �V ; " * " / � " / � / V < ; V� ; VV " > > > D ) > � ) > � , �( " / V � (C D , N " / � � (& /, '̂, � " / � � (� � )

Results – Eigenvalue and eigenmodes �� � 80 � � � 0.3 � � 50 W D � 0.57 Eigenvalue spectrum Unstables eigenmodes 9 � 0.85 ? Fluid-solid (short-circuit) 9 � 1.33 ? Fluid-solid-electric (open-circuit) Maximal displacement increased by 3 orders of magnitude !

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

Free-vibration modes – Frequency comparison Z � <. \] Fluid-solid (short-circuit) Free-vibration modes of the elastic plate Z [ � <. \] 9 ? � 5.25 9 _ � 14.59 Fluid-solid-electric (open-circuit) Z � [. aa Free-vibration modes of the elastic plate with piezo-patches Z [ � [. aa 9 ? � 7.64 9 _ � 17.66

Free-vibration modes – Amplitude comparison Fluid-solid (short-circuit) Free-vibration modes of the elastic plate c [ � <. dded f ? � 0.0029 f _ � 0.0002 Fluid-solid-electric (open-circuit) Free-vibration modes of the elastic plate with piezo-patches c [ � <. ddgd f ? � 0.0017 f _ � 0.0004

Results for the piezo R-shunt configuration Behaviour of the leading eigenvalue when varying the resistance Growth rate Frequency Stabilization in a range Jump of the frequency of electric resistance

Results - Piezo-shunt configuration Behaviour of the leading eigenvalue when varying the resistance Eigenvalue spectrum Short-circuit Open-circuit � D � 0 � D → ∞ Small h V : stabilization of the fluid-solid eigenmode Large h V : destabilization of the fluid-solid-electric eigenmode

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) - Active feeback control