Large-Scale Numerical Simulation of Fluid Structure Interactions in - PowerPoint PPT Presentation

Large-Scale Numerical Simulation of Fluid Structure Interactions in Low Reynolds Number Flows APS 64 rd Annual Meeting Division of Fluid Dynamics Ali EKEN & Mehmet SAHIN 20 November 2011 — Baltimore, Maryland, USA Astronautical Engineering Department, Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469, Maslak/Isatanbul, TURKEY

CONTENTS • Motivations • Numerical Modelling • Structure Solver • Structure Solver Validation • Test Case I: Plate with a Hole • ALE Fluid Solver • ALE Fluid Solver Validation • Test Case II: The Flow Past an Oscillating Circular Cylinder in a Channel • Fluid Structure Coupling • Fluid Syructure Solver Validation • Test Case III: Vortex-induced vibration of an elastic cantilever beam • Test Case IV: 3-D Elastic Solid in a Steady Channel Flow • Conslutions and Future Works

Motivations Fluid structure interactions for a 3d parachute Membrane wing fluid structure interaction Trimarchi at al., (2011). Stanford at al., (2008). Fighter tail buffeting and aircraft flutter (from NASA). Vascular fluid structure interaction Bazilevs at al., (2010).

Structure Solver

Structure Solver Equations of Solid Motion The governing equations of motion within the solid domain: Galerkin Finite Element Formulation: Displacement Boundary Conditions: Traction Boundary Conditions:

Structure Solver Finite Element Discretization with Incompatible Modes

Structure Solver Finite Element Discretization with Incompatible Modes The sti ff ness Matrix: Mass Matrix: Surface Tractions: Static Condensation:

Structure Solver Solution of Resulting System of Equations The system of equations: Newmark Method:

Structure Solver Test Case I: Plate with a Hole 5N of concentrated loads at the free end Dimensions 100x100x1 mm Hole diameter 60 mm

Structure Solver Test Case I: Plate with a Hole: Displacement Results

ALE Fluid Solver

ALE Solver Equations of Fluid Motion The integral form of the incompressible Navier-Stokes equations on deforming meshes: The momentum equation: The continuity equation: No-slip wall boundary condition: Inflow boundary condition: Outflow boundary condition:

Numerical Discretization (a) Two-dimensional dual volume (b) Three-dimensional dual volume The side centered finite volume was initially used by Hwang (1995) and Rida et al. (1997) on unstructured meshes. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance pressure-velocity coupling. The most appealing feature of this primitive variable arrangement is the availability of very efficient multigrid solvers.

ALE Solver Numerical Discretization (Continued...) The computed contributions for the x -momentum equation are given below for the right element The time derivation: The convective term due to fluid motion:

ALE Solver Numerical Discretization (Continued...) The Geometric Conservaion Law (GCL) is satisfied The convective term due to grid motion: The pressure term:

ALE Solver Numerical Discretization (Continued...) The viscous term: The gradient terms are calculated from the Gauss-Green theorem:

ALE Solver TEST CASE II: An Oscillating Circular Cylinder in a Channel The kinematic viscosity Density The cylinder center is oscillating sinusoidally such that the location of the cylinder center is given by and , . The computational mesh consists of 70667 quadrilateral elements and the time step is set to .

ALE Solver TEST CASE II: An Oscillating Circular Cylinder in a Channel t=20.0s t=21.0s t=22.0s t=23.0s u -velocity vector component contours with streamtraces for an oscillating circular cylinder in a channel at several different time levels. .

ALE Solver TEST CASE II: An Oscillating Circular Cylinder in a Channel The comparison of the c d and c l with the numerical results of Wan and Turek (JCP , 2007).

Fluid Structure Interaction

FSI Solver Fluid Structure Interaction Coupling Fluid domain: Solid domain: Fluid-structure interface: Fluid on rigid boundary:

FSI Solver The Fully Coupled System The resulting algebraic linear system: The modified system: Three banded matrix Additive Schwartz preconditioned GMRES(m) algorithm is used to solve the algebraic equations. An ILU(4) preconditioner with rcm ordering is employed within each partioned subdomains. The implementation of the Krylov subspace methods, preconditioning and matrix-matrix multiplication have been caried out using PETSc library. In order to encounter non-linearity due to uknown vertex locations at (n+1) several sub iterations are performed.

FSI Solver Parallelization and Efficiency Mesh Generation (GAMBIT, CUBIT, ...) Mesh Partition (METIS library) Parallel Fully-Coupled FSI Code Linear Solver Kroylov subspace methods (PETSc library) Preconditioners (PETSc library) Post Processing (Tecplot)

FSI Solver TEST CASE III: Vortex-induced vibration of an elastic plate Fluid: Solid:

FSI Solver TEST CASE III: Vortex-induced vibration of an elastic plate Coarse mesh with 40935 quadrilateral elements and 41395 nodes (273 185 DOF). Generated using CUBIT library.

FSI Solver TEST CASE III: Vortex-induced vibration of an elastic plate Fine mesh with 125443 quadrilateral elements and 126920 nodes (837 609 DOF). Generated using CUBIT library.

FSI Solver TEST CASE III: Vortex-induced vibration of an elastic plate Due to large startup vortex Periodic state The time variation of the computed tip displacement for the coarse and fine meshes. The time step is set to 0.001 and Re=332.60.

FSI Solver TEST CASE III: Vortex-induced vibration of an elastic plate The instantaneous u -velocity contours with streamlines at t=3.0.

FSI Solver TEST CASE III: Vortex-induced vibration of an elastic plate The instantaneous vorticity contours with streamlines at t=3.0.

FSI Solver TEST CASE IV: 3D elastic solid in a steady channel flow Fluid: Solid: Richter, (2011)

FSI Solver TEST CASE IV: 3D elastic solid in a steady channel flow The computational mesh with local mesh refinement using CUBIT library. The half of the doamin is shown. The mesh consists of 374523 nodes and 362224 hexahedral elements (4 096 651 DOF).

FSI Solver TEST CASE IV: 3D elastic solid in a steady channel flow The computed streamtraces at Re=40. The color shows the magnitude of the u -velocity component.

FSI Solver TEST CASE IV: 3D elastic solid in a steady channel flow The computed isobaric surfaces at Re=40. The color shows the magnitude of the pressure value.

FSI Solver TEST CASE IV: 3D elastic solid in a steady channel flow St. Venant Kirchhoff material with large displacement The computed displacement vectors at Re=40. The color shows the magnitude of the x -displacement.

FSI Solver Conclusions and Future Works A parallel unstructured fluid structure interaction (FSI) code  has been developed. The fluid solver is based on the solution of the incompressible  Navier-Stokes equations using an ALE formulation. The solid solver is based on the linear elastic material model  with small deformations. The large deformations with non- linear models will be added to the present code. Currently we use one level preconditioned iterative solver for  the whole coupled solver. In the future, the fully coupled sytem will be solved using monolitic preconditioners. The present code will be applied to solve the fluid structure  interactions for memrane wing Micro Aerial Vehicles (MAV).

Acknowledgement The authors gratefully acknowledge the use of the Chimera machine at the Faculty of Aeronautics and Astronautics at ITU, the computing resources provided by the National Center for High Performance Computing of Turkey (UYBHM) under grant number 10752009 and the computing facilities at TUBITAK ULAKBIM, High Performance and Grid Computing Center. The authors also would like to thank Thomas Richter at the University of Heidelberg for performing the validation cases.