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
Introduction Fluid-Structure system exhibit various types of instability phenomenons 2
Introduction Fluid-Structure system exhibit various types of instability phenomenons � Accurate fluid-structure modeling is needed 3
Introduction Turbulent flows Streamlined body - Attached flow Bluff body - Separated flow Heaving and pitching involved Only pitching motion involved Potential flow modelling No robust modelling Analytical methods [Theodorsen 1935] Experiments / Time-marching simulations in [Païdoussis 2011] 4
Introduction Turbulent flows Streamlined body - Attached flow Bluff body - Separated flow Heaving and pitching involved Only pitching motion involved Potential flow modelling No robust modelling Analytical methods [Theodorsen 1935] Experiments / Time-marching simulations Proposed method Linear stability analysis of the coupled fluid-structure interaction problem with a turbulent flow modelled with Reynolds Averaged Navier Stokes (RANS) equations 5
Part 1 – Configuration and Modelling Part 2 – An elongated plate mounted on two springs Part 3 – A short plate mounted on two springs 6
Part 1 : Configuration and Modelling 7
Aeroelastic configuration Heaving and pitching rigid body in turbulent incompressible flows Elastic axis � � , � , � � + � �� Mass center � � � ℎ � � Torsional spring Translational spring 8
Aeroelastic configuration � �� = � � � � � Two aspect ratios are investigated Short plate Elongated plate �� = � � 23 5 « Bridge type » « Airfoil type » 9
Modelisation Non-dimensional numbers : �� = �� � Fluid parameter : � " #$ : non-dimensional ! = � �� heaving natural � �� frequency Solid parameter : " #% : non-dimensional 1 � ∗ = pitching natural � �� frequency = � � � Coupling parameter : � � � = � �� �� = � Shape parameters : � � 10 10
Modelisation Non-dimensional numbers : �� = 2,7 ⋅ 10 + Fluid parameter : ! = 0,8 Solid parameter : 1 � ∗ = � �� = 10 + � Coupling parameter : �� = � � = 0,08 Shape parameters : � 11 11
Fluid-structure modelling Fluid variables Solid variables - 2 - . ,- . Fluid equation ,/ = 0 . - . + 1 .2 (- . , - 2 ) 5- 2 Solid equation 5/ = 1 2. - . + 0 2 - 2 Coupled fluid-structure equations 12 12
Fluid-structure modelling Fluid dynamics � RANS approach � Spalart-Allmaras turbulent model - . = 6, 7, 8, �̃ : ,- . ,/ = 0 . - . + 1 .2 (- . , - 2 ) 5- 2 5/ = 1 2. - . + 0 2 - 2 13 13
Fluid-structure modelling Fluid dynamics � RANS approach � Spalart-Allmaras turbulent model - . = 6, 7, 8, �̃ : ,- . ,/ = 0 . - . + 1 .2 (- . , - 2 ) 5- 2 5/ = 1 2. - . + 0 2 - 2 - 2 = ℎ, �, ℎ>, �> : Solid dynamics � 2-DOF rigid body < ℎ + � �; = 0 ℎ; + � �� < � + =(�) ℎ; = 0 �; + � �� 14 14
Fluid-structure modelling Fluid dynamics Solid-to-fluid coupling � RANS approach � Interface conditions � Spalart-Allmaras turbulent model � Non-inertial volumic terms [Mougin et al. 2002] - . = 6, 7, 8, �̃ : ,- . ,/ = 0 . - . + 1 .2 (- . , - 2 ) 5- 2 5/ = 1 2. - . + 0 2 - 2 - 2 = ℎ, �, ℎ>, �> : Solid dynamics Fluid-to-solid coupling � 2-DOF rigid body � Fluid force - Lift < ℎ + � �; = 0 ℎ; + � �� � Fluid moment < � + =(�) ℎ; = 0 �; + � �� 15 15
Linear stability analysis Two classical ingredients : Perturbation decomposition B ?, / - . ?, / = @ . ? + A - . Fluid B / Solid - 2 / = # + A - 2 base state perturbation Modal decomposition B ?, C = - D . ? E FG + H. H - . Fluid B C = - D 2 E FG + H. H Solid - 2 16 16
Linear stability analysis Growth rate Frequency Fluid-solid mode J = ℜ[M] D . , - D 2 ) � = ℑ[M] (- Coupled eigenvalue problem = � �� � �� M P - D . - D . � �� � �� D 2 D 2 - - Mass matrix Fluid-structure (spatial discretizarion) Jacobian matrix 17 17
Part 2 : An elongated plate mounted on two springs (AR=23) 18
Steady base flow Axial velocity ( �� = 27500 ) Turbulent to kinematic viscosity ratio 19 19
Steady base flow 20 20
Steady base flow Lift coefficient Moment coefficient ,Q R ,Q W ,S T ,S T UV� UV� Present study 7,0 1,8 Potential flows 7,0 1,7 21 21
Steady base flow Lift coefficient Moment coefficient ,Q R ,Q W ,S T ,S T Small influence of UV� UV� detached areas on Present study 7,0 1,8 slopes at S = 0° Potential flows 7,0 1,7 22 22
Linear Stability Analysis Z .. Z .2 MY- D = Z- D Z = Coupled fluid-structure system Z 2. Z 22 23 23
Linear Stability Analysis D . = Z �� - D . M � Y .. - Uncoupled fluid system Fluid system spectrum 24 24
Linear Stability Analysis D . = Z �� - D . M � Y .. - Uncoupled fluid system Fluid system spectrum 25 25
Linear Stability Analysis D . = Z �� - D . M � Y .. - Uncoupled fluid system � = 20,7 ( [/ \ ≃ 0,14 ) Fluid system spectrum 26 26
Linear Stability Analysis D . = Z �� - D . M � Y .. - Uncoupled fluid system Uncoupled Fluid spectrum � = 20,7 ( [/ \ ≃ 0,14 ) Fluid system spectrum 27 27
Linear Stability Analysis M � Y 22 - D 2 = Z �� - D 2 Uncoupled solid system Low-frequency mode High-frequency mode Uncoupled Solid spectrum Uncoupled Fluid spectrum � � = 1,08 = 0,76 � �� � �� � ∗ = 5 � Inertial coupling bewteen heaving and pitching 28 28
Linear Stability Analysis Uncoupled fluid and solid systems 29 29
Linear Stability Analysis Coupled fluid-structure system 30 30
Linear Stability Analysis Z .. Z .2 MY- D = Z- D Z = Coupled fluid-structure system Z 2. Z 22 � ∗ = 5 = 10 + � 31 31
Linear Stability Analysis Z .. Z .2 M- D = Z- D Z = Coupled fluid-structure system Z 2. Z 22 � = 10 a Mode kinetic energy : ` � = 1 ` 32 32
Linear Stability Analysis Z .. Z .2 M- D = Z- D Z = Coupled fluid-structure system Z 2. Z 22 Mode kinetic energy : ` � = 1 ` � = 4,1 33 33
Linear Stability Analysis Z .. Z .2 M- D = Z- D Z = Coupled fluid-structure system Z 2. Z 22 Mode kinetic energy : ` � = 1 ` � = 1,1 34 34
Aeroelastic instabilities of a 2-DOF elongated plate � ∗ → 0 : uncoupled case 35
Aeroelastic instabilities of a 2-DOF elongated plate Coupled mode flutter 36
Aeroelastic instabilities of a 2-DOF elongated plate Coupled mode flutter � ∗ = 5 37
Aeroelastic instabilities of a 2-DOF elongated plate VIV (not investigated here) Coupled mode flutter 1 � ∗ = 5 ∗ � cdc ≃ 2e [/ \ �� ≃ 0,05 38
Validation � ∗ � fghG � �� Present study 2,9 0,81 Theodorsen 3,0 0,81 Comparison to Theodorsen model Modelisation validated against � classical Theodorsen flutter theory Maginal role of detached areas on � instability thresholds 39
Part 3 : An short plate mounted on two springs (AR=5) 40
Steady Base Flow 0,75 � ,Q R ,Q W Large leading-edge detached ,S T ,S T areas UV� UV� � Non-negligible effect on Present study 9,15 0,95 steady aerodynamic Potential flows 8,4 1,7 coefficient 41
Aeroelastic instabilities of a 2-DOF short plate 42
Aeroelastic instabilities of a 2-DOF short plate VIV (not investigated here) 1 ∗ � cdc ≃ 2e [/ \ �� ≃ 0,2 43
Aeroelastic instabilities of a 2-DOF short plate VIV (not investigated here) Single mode high frequency flutter 1 ∗ � cdc ≃ 2e [/ \ �� ≃ 0,2 44
Aeroelastic instabilities of a 2-DOF short plate Single mode high frequency flutter 45
Aeroelastic instabilities of a 2-DOF short plate Single mode low frequency flutter Single mode high frequency flutter 46
Aeroelastic instabilities of a 2-DOF short plate Single mode low frequency flutter Single mode high frequency flutter 47
Aeroelastic instabilities of a 2-DOF short plate Low frequency flutter mode 48
Aeroelastic instabilities of a 2-DOF short plate Single mode low frequency flutter Single mode high frequency flutter 49
Aeroelastic instabilities of a 2-DOF short plate Low frequency flutter mode High frequency flutter mode 50
Comparison to Theodorsen … ∗ "/" #% i jklC Present study 0,98 0,77 (low frequency branch) Present study ~1,4 1,08 (high frequency branch) Theodorsen 4,3 1,03 Clear difference bewteen RANS and Theodorsen o Well known phenomena : single-mode flutter is difficult to model o Neither Theodorsen nor other simple method work � � Interest of using a full RANS modelling 51
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