ITU/FAA Faculty of Aeronautics and Astronautics An ALE Approach for the Numerical Simulation of Drosophila Flight Belkis ERZINCANLI & Mehmet SAHIN 6 th European Congress on Computational Methods in Applied Sciences and Engineering, September 10-14, 2012 – Vienna, Austria. Aeronautical Engineering Department, Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469, Maslak/Isatanbul, TURKEY
ITU/FAA Faculty of Aeronautics and Astronautics Contents • Motivations • ALE Formulation • Governing equations • Discrete Geometric Conservation Law (GCL) • 3D Numerical discretization • Iterative solver • Numerical Validations • Kovasznay flow • The flow past an oscillating circular cylinder in a channel • The flow induced due to a horizontally oscillating sphere • The Numerical Simulation of Flow Field Around Drosophila • Conclusions and Future Work ECCOMAS 2012 2
ITU/FAA Faculty of Aeronautics and Astronautics The Fruit Fly , Drosophila Debat et al., Evolution, 60(12), 2006, pp. 2529 – 2538 ECCOMAS 2012 3
ITU/FAA Faculty of Aeronautics and Astronautics Insect Flight Aerodynamics A comparison of 2-D linear translation vs 3-D Clap & Fling flapping translation S ane, S.P.,”The Aerodynamics of Insect Flight”, The Journal of Experimental Biology 206,4191-4208, (2003). Stable attachment of the leading edge vortex ECCOMAS 2012 4
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Aprroaches Used for Moving Boundary Flow Problems • Immersed Boundary Method (IBM) : This approach solves Navier-Stokes equations on a Cartesian mesh with a moving immersed body (Miller & Peskin, 2004; Gilmanov & Sotiropoulos, 2005; Mittal et al., 2006; Bush et al., ) . • Fictious Domain Method : This approach uses Eulerian mesh and the body motion is represented using the distributed Lagrangian multiplier method. • Overset (Chimera) Grid Method : This method uses several body fitted meshes inside the fluid domain and an interpolation algorithm is used between the neighbouring meshes (Liu and Aono, 2009). • Arbitrary Lagrangian-Eulerian (ALE) Method : The mesh follows the interface between the fluid and solid boundary and the governing equations are discretized on a moving/deforming mesh (Johnson, 2006; Ramamurti & Sandberg, 2002). ECCOMAS 2012 5
ITU/FAA Faculty of Aeronautics and Astronautics A Fully Coupled Parallel Unstructured Finite Volume Method based on ALE Formulation • In the Arbitrary Lagrangian-Eulerian (ALE) method, the interface between the solid and fluid is sharply represented. • The mesh movement has to satisfy a special condition called Geometric Conservation Law (GCL) in order to maintain the accuracy and the stability of the time integration scheme. • An unstructured ALE method based on the side-centered finite volume technique has been developed for parallel large-scale moving boundary problems. • 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 the pressure-velocity coupling. • The most appealing feature of present primitive variable arrangement is the availability of very efficient multigrid solvers. • A fully coupled implicit iterative solver is used for large-scale calculations since the CFL restriction may lead to inadmissible time steps on deforming meshes and does not introduce any splitting error when the advection- diffusion operator is treated impicitly. ECCOMAS 2012 6
ITU/FAA Faculty of Aeronautics and Astronautics Mathematical and Numerical Formulation The incompressible Navier-Stokes equations that govern the viscous fluid flow of an arbitrary moving Eulerian Lagrangian control volume Ω (t) with boundary ∂Ω (t) can be written in Cartesian coordinate system in dimensionless form as follows: The continuity equation : The momentum equation : Three-dimensional dual volume Grid velocity ECCOMAS 2012 7
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Side-centered finite volume method • A stable numerical scheme with exact mass conservation • No ad-hoc modifications for pressure-velocity coupling • Very efficient multigrid solvers are available (a) Two-dimensional dual volume (b) Three-dimensional dual volume ECCOMAS 2012 8
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The contributions for the x -momentum equation for the right element: The time derivation is the volume of the pyramid between points ECCOMAS 2012 9
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The convective term due to fluid motion are the area vectors of the dual volume triangular surfaces. are the velocity vectors defined at the mid- point of each dual volume surface area. ECCOMAS 2012 10
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The convective term due to mesh motion First-order backward difference is used for grid velocity Geometric conservation law satisfied. ECCOMAS 2012 11
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The pressure term 2nd order Taylor series expansion is used for the pressure values at . ECCOMAS 2012 12
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The viscous term Gauss-Green theorem is used for the gradient terms: ECCOMAS 2012 13
ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The continuity equation is integrated within each hexahedral elements and evaluated using the mid-point rule on each of the element faces where is the hexahedral element surface area vector and u , v and w are the velocity vector components defined at the mid-point of each hexahedral element face. The discretization of above equations leads to a saddle point problem of the form: where, are the convection diffusion operators, is the pressure gradient operator and is the divergence operator. ECCOMAS 2012 14
ITU/FAA Faculty of Aeronautics and Astronautics Iterative Method The preconditioner matrix is Where .For the inverse of the scaled Laplacian S, we use two-cycle AMG solver provided by the HYPRE library, a high performance preconditioning package developed at Lawrence Livermore National Laboratory, which we access through the PETSC library. ECCOMAS 2012 15
ITU/FAA Faculty of Aeronautics and Astronautics Parallelization and Efficiency Mesh Generation (GAMBIT, CUBIT, ...) Mesh Partition (METIS library) Parallel Unstructured ALE Code Linear Solver with Multigrid Preconditioner Kroylov subspace methods (PETSc library) Preconditoners (PETSc and HYPRE libraries ) Post Processing (Tecplot) ECCOMAS 2012 16
ITU/FAA Faculty of Aeronautics and Astronautics Computing Resources for Parallel Calculations • SGI Altix 3000 (1300MHz, Itanium 2) with 32 nodes • National Center for High Performance Computing of Turkey • TUBITAK ULAKBIM, High Performance and Grid Computing Center ECCOMAS 2012 17
ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE I: 2D Kovasznay Flow The spatial convergence of the error with mesh refinement . ECCOMAS 2012 18
ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: An Oscillating Circular Cylinder in a Channel • The location of the cylinder center • Amplitude and frequency of the oscillation • Time step • Reynolds number • 70,667 quadrilateral elements • 71,349 vertices • 354,699 total DOF ECCOMAS 2012 19
ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: An Oscillating Circular Cylinder in a Channel t=20.0s t=21.0s t=22.0s t=23.0s The computed u -velocity vector component contours with streamtraces. ECCOMAS 2012 20
ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: An Oscillating Circular Cylinder in a Channel The comparion of c d and c l plots with results from Wan and Turek (JCP, 2007) . ECCOMAS 2012 21
ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE III: An Oscillating Sphere in a Cubic Cavity • The location of the sphere • Time step • Reynolds number (Gilmanov & Sotiropoulos , 2005) The Reynolds number is based on the sphere diameter and the maximum sphere velocity. • 473,856 hexahedral elements • 492,596 vertices • 4,794,264 total DOF ECCOMAS 2012 22
ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE III: An Oscillating Sphere in a Cubic Cavity • The location of the sphere x ( t ) = h [1- cos (2 t )] h = 0.125 D The instantaneous streamtraces with u- velocity component contours at different phases over a cycle for a rigid sphere oscillating in a cubic cavity at Re = 20: t=0, t=T/4, t=T/2 and t=3T/4 . The time variation of drag coefficient for a rigid sphere oscillating in a cubic cavity at Re = 20. ECCOMAS 2012 23
ITU/FAA Faculty of Aeronautics and Astronautics The Numerical Simulation of Flow Field Around Drosophila Robofly experiment of M. Dickinson (CalTech / ICB), Science (1999). ECCOMAS 2012 24
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