Assessment of Fictitious Domain method for Linear Stability Analysis - PowerPoint PPT Presentation

Assessment of Fictitious Domain method for Linear Stability Analysis of Fluid-Structure systems J. Moulin, J-L. Pfister, M.Carini, O.Marquet Office National d’Etudes et de Recherches Aérospatiales, Département Aérodynamique Fondamentale et Expérimentale Funded by ERC Starting Grant ERCOFTAC-SIG33, Siena, 19-21 June 2017

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Linear Stability Analysis & FSI Solid modelling complexity Fluttering flag (C.Eloy, JFM 2008) Potential flow + 1D elastic solid Spring-Mounted airfoil Zig-Zag mode (J.Tchoufag, JFM 2014) Quasi-Steady theory + damped oscillator Navier-Stokes + rigid-solid Fluid modelling complexity 2/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Linear Stability Analysis & FSI Solid modelling complexity Fluttering flag (C.Eloy, JFM 2008) Potential flow + 1D elastic solid Spring-Mounted airfoil Zig-Zag mode (J.Tchoufag, JFM 2014) Quasi-Steady theory + damped oscillator Navier-Stokes + rigid-solid Fluid modelling complexity 2/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Linear Stability Analysis & FSI Solid modelling complexity Fluttering flag (C.Eloy, JFM 2008) Potential flow + 1D elastic solid Spring-Mounted airfoil Zig-Zag mode (J.Tchoufag, JFM 2014) Quasi-Steady theory + damped oscillator Navier-Stokes + rigid-solid Fluid modelling complexity Objective Assess the use of Fictitious Domain approach for Stability Analysis of FSI systems 2/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Plan Introduction 1 Conforming vs. Non-Conforming 2 App.1 : VIV on rigid cylinder 3 App. 2 : cylinder with flexible appendice 4 Conclusion 5 3/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Numerical frameworks for FSI Description of the separate problems Ω s ( t ) u Fluid : Eulerian description in domain Ω f ( t ) Solid : Lagrangian description ξ s in domain Ω s Ω f ( t ) 0 4/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Numerical frameworks for FSI Description of the separate problems Ω s ( t ) u Fluid : Eulerian description in domain Ω f ( t ) Solid : Lagrangian description ξ s in domain Ω s Ω f ( t ) 0 Description of coupled FSI problems Conforming methods Non-Conforming methods Arbitrary Lagrangian-Eulerian method (ALE) Fictitious Domain method (FD) (J.Donea, Enc. Comp. Mech. 2004) (R.Glowinsky, Int. J. Numer. Meth. Fluid 1999) Artificial unknowns : mesh displacement in Artificial unknowns : fluid velocity in the ➥ ➥ the fluid region solid region Added equation : fluid mesh movement Added equation : constraint equation to ➥ ➥ equation impose the solid presence 5/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Numerical frameworks for FSI Description of the separate problems Ω s ( t ) u Fluid : Eulerian description in domain Ω f ( t ) Solid : Lagrangian description ξ s in domain Ω s Ω f ( t ) 0 Description of coupled FSI problems Conforming methods (Ref.) Non-Conforming methods Arbitrary Lagrangian-Eulerian method (ALE) Fictitious Domain method (FD) (J.Donea, Enc. Comp. Mech. 2004) (R.Glowinsky, Int. J. Numer. Meth. Fluid 1999) Artificial unknowns : mesh displacement in Artificial unknowns : fluid velocity in the ➥ ➥ the fluid region solid region Added equation : fluid mesh movement Added equation : constraint equation to ➥ ➥ equation impose the solid presence 6/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion An illustrative example Heat equation in a moving domain Ω f ( ξ s ( t )) : ∆ T = 0 in Ω f ( ξ s ( t )) T = 1 ξ s ( t ) T = 0 T = 1 T = 1 Ω f T = 1 7/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion An illustrative example Heat equation in a moving domain Ω f ( ξ s ( t )) : ∆ T = 0 in Ω f ( ξ s ( t )) T = 1 ξ s ( t ) T = 0 T = 1 T = 1 Ω f T = 1 How is the fluid-structure coupling handled in ALE vs. Fictitious Domain frameworks ? 7/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion ALE formalism : an illustrative example Current configuration x Ω f ( ξ s ) 8/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion ALE formalism : an illustrative example Current configuration x Ω f ( ξ s ) � ∇ T · ∇ v d x = 0 Ω f ( ξ s ) 8/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion ALE formalism : an illustrative example Current configuration Reference configuration x = x r + ξ ( x r , t ) F = ∂ x ∂ x r J = det ( F ) x x r Ω f Ω f ( ξ s ) 0 � ∇ T · ∇ v d x = 0 Ω f ( ξ s ) 8/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion ALE formalism : an illustrative example Current configuration Reference configuration x = x r + ξ ( x r , t ) F = ∂ x ∂ x r J = det ( F ) x x r Ω f Ω f ( ξ s ) 0 � � ( F ( ξ s ) − T ∇ T ) · ( F ( ξ s ) − T ∇ v ) J ( ξ s ) d x r = 0 ∇ T · ∇ v d x = 0 Ω f Ω f ( ξ s ) 0 8/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion ALE formalism : an illustrative example Current configuration Reference configuration x = x r + ξ ( x r , t ) F = ∂ x ∂ x r J = det ( F ) x x r Ω f Ω f ( ξ s ) 0 � � ( F ( ξ s ) − T ∇ T ) · ( F ( ξ s ) − T ∇ v ) J ( ξ s ) d x r = 0 ∇ T · ∇ v d x = 0 Ω f Ω f ( ξ s ) 0 The geometrical non-linearity can be put either : in the fluid integration domain or in the fluid integrand 8/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion ALE formalism : an illustrative example Current configuration Reference configuration x = x r + ξ ( x r , t ) F = ∂ x ∂ x r J = det ( F ) x x r Ω f Ω f ( ξ s ) 0 � � ( F ( ξ s ) − T ∇ T ) · ( F ( ξ s ) − T ∇ v ) J ( ξ s ) d x r = 0 ∇ T · ∇ v d x = 0 Ω f Ω f ( ξ s ) 0 The geometrical non-linearity can be put either : in the fluid integration domain or in the fluid integrand Imagine what will happen with full Navier-Stokes ... 8/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Fictitious Domain formalism : an illustrative example Current configuration Ω 9/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems

Introduction Conforming vs. Non-Conforming App.1 : VIV on rigid cylinder App. 2 : cylinder with flexible appendice Conclusion Fictitious Domain formalism : an illustrative example Current configuration x Ω s ( ξ s ) Ω 9/22 J.Moulin - ERCOFTAC-SIG33, Siena, 19-21 June 2017 - Linear Stability Analysis of Fluid-Structure systems