10 th Symposimum on Overset Composite Grids and Solution Technology, NASA Ames Research Center Moffett Field, California, USA 1 Implementing a Partitioned Algorithm for Fluid-Structure Interaction of Flexible Flapping Wings within Overture Dominic D J Chandar Postdoctoral Research Associate University of Wyoming, USA Murali Damodaran Professor, Indian Institute of Technology, Gandhinagar, India Work Done at Nanyang Technological University Singapore Sep 21 2010
2 Outline Overture Framework Definition of various problems Computational Framework Key results and discussions Concluding remarks Overlapping grids Fluid dynamics Structural dynamics Rigid body dynamics
3 Overture Framework Overture / OverBlown (CGINS) Codes Overset Moving Grids (ogen) Deforming Grid 3D + Improvements to Numerical Stability Parallel Computing on Stationary Grids Open-Source Overture Framework (LLNL Codes) A++ Framework Operator Overloading Overset Flow Solver (OverBlown/cgins) Interface With PETSc FSI CSD Solver (cgsd)
4 Summary of Work Undertaken using Overture Rigid and flexible flapping airfoils and wings Wing deformation & Fluid Structure Interaction Effect of flexure on Aerodynamic characteristics Passive Flight Vortex dynamics & Thrust Generation Dynamic Stall & Lift Hysteresis Lift Decreases with increase in angle of attack How surrounding vorticity field Influences thrust Accelerating motion of a flapping wing due to Aerodynamic forces
5 Overlapping / Composite / Overset / Chimera Grids Individual grids can be Generated independently Ogen Mesh Generator Interpolation
6 Moving Overlapping Grids Two-Dimensional Moving Grid (Rigid) Three-Dimensional Deforming Grid
7 Computational Flow Modeling (A) Fluid Dynamics Incompressible Navier Stokes Equations u Velocity Vector p Static pressure ρ Fluid density Pressure Poisson Equation 2nd Order spatial differences 2nd Order Crank Nicolson Implicit (For Viscous terms) 2nd Order Adams Predictor-Corrector (Explicit )
8 Computational Flow Modeling
9 Computational Modeling (B) 6 DOF Rigid Body Dynamics e2 FA= Net Aerodynamic Force T = Net Aerodynamic Torque / Moment about Centre of mass ( xcms) e1 T M = Mass of the body I = Components of Principal moments of Inertia ω= Angular velocity vector y ei = Principal axes. τ = Stress Tensor FA x Position Orientation
10 Computational Fluid-Structure Interaction and Coupling Issues WING Rigid body motion of the wing Flapping Aerodynamic Forces Motion Wing deformation due to aerodynamic forces Air flowing over the wing
11 Computational Structural Dynamics Modeling Structural Dynamics Kirchhoff s Plate Equation 3D mxz wtt + D w = q ( x, z, t ) 4 Eh3 D= 12 ( 1 ν Euler-Bernoulli Beam Equation m Vertical Oscillations Rigid 2w x t 2 2 ) + EI 4w x 4 = q ( x, t ) Flexible E : Modulus of Elasticity, h : Plate Thickness, ν : Poissons Ratio, mxz : Mass per unit area, q: Load Acting, I: Moment of Inertia 2D
12 Computational Structural Dynamics Modeling Structural Dynamics x z Governing Eqn mxz wtt + D 4 w = q ( x, z, t ) Static Part w= 4 13 i= 1 Dynamic Part κ w& Ai wi = [Q]w, Ai = A, B...M n + 1 t 2 q ( x, z, t ) n + 1 t2d [Q] w = + 2 wn wn 1 I+ mxz mxz
13 Computational Structural Dynamics Modeling Discretization of the Euler-Bernoulli Beam Equation : m win + 1 2 win + win 1 x t2 Solution + EI [ A] y n+ 1 win++21 4 win++11 + 6 win + 1 4 win +11 + win +21 x4 n = f (y,q n+ 1 ) 2 O( t, x ) = q ( x, t ) n+ 1
14 Fluid Structure Coupling Partitioned Approach (Dirichlet Neumann Approach) With Inner Iterations : Time = Tn Transfer stresses through interpolation Time = Tn+1
15 Fluid Structure Coupling Load Transfer from Fluid to Solid: Interpolate Wing Caps on to Wing STEP 1
16 Fluid Structure Coupling Interpolate CSD Grid from CFD Stress Transfer from Fluid to Solid: (Upper) (Lower) 16
17 Fluid Structure Coupling Comparison of Actual and Interpolated Stress Actual (CFD) Interpolated (CSD) Upper Surface Lower Surface 17
18 Fluid Structure Coupling Partitioned Approach (Dirichlet Neumann Approach) With Inner Iterations : Time = Tn Transfer stresses through interpolation Transfer displacements through interpolation Local Iterations Time = Tn+1 How to choose ω
19 Computational Cases Investigated Rigid Plunging Wing Plunging and (active) deforming airfoil Plunging and (passive) deforming airfoil Deformation of a beam in a fluid FSI Coupling Issues Plunging and passively deforming wing
20 Rigid Plunging Wing Comparison of Aerodynamic forces and moments Experiment Navier-Stokes Panel Method Linear Theory Heathcote et al. (2008) Young (2005) Young (2005) Garrick (1936) Motion along Y-axis is given by : Reduced frequency, k = 0.5, 1.0, 1.82, 2.5, 3.5 and 4.0 Reynolds number, Re = 104
21 Rigid Plunging Wing Thrust Coefficient Power Input Coefficient Propulsive Efficiency
22 Plunging and Deforming (Active) Airfoil Y NACA 0014 Airfoil X Vertical Oscillations Kinematics : + Chordwise deformation STATIC: RIGID BODY MOTION: RIGID BODY + DEFORMATION: 1. Rigid Plunging : B = 0.4, k=2, Re=104, A =0 - Tuncer and Kaya (2003) 2. Plunging with Deformation : A = 0.3, p = 2, xc=0, Q=0, φ=0, Miao and Ho (2006)
23 Plunging and Deforming (Active) Airfoil 1. Rigid Plunging : B = 0.4, k=2, Re=104, A =0 - Tuncer and Kaya (2003)
24 Plunging and Deforming (Active) Airfoil 2. Plunging with Deformation : A = 0.3, p = 2, xc=0, Q=0, φ=0, Miao and Ho (2006)
25 Plunging and Deforming (Active) Airfoil Miao & Ho (2006) Present computation Cp Contours at the mean position
26 Plunging and Deforming (Passive) Airfoil Heathcote et al. (2004) Tang et al. (2007) Rigid : Length 0.4c, Thickness 0.11c Vertical Oscillations Rigid Flexible Flex : Length 0.6c, Thickness 0.005c Re = 9000 E = 2.05x1010ρU2 k = fc/u = 1.4 ρb = 7.85 ρ Mesh Size : 500 x 11, 200 x 150 k = reduced frequency, f = frequency, c = beam chord, ρb= beam density, ρ = Fluid Density, U = FreeStream Velocity, E = Modulus of Elasticity
27 Effect of Time Step, Relaxation and Damping Tip Displacement With Net Iteration Index for t = 1e-3 STABLE UNSTABLE Tip Displacement STABLE time t=0 t=dt t=2dt Rigid : Length 0.4c, Thickness 0.11c Flex : Length 0.6c, Thickness 0.005c Re = 9000 E = 2.05x1010ρU2 k = fc/u = 1.4 ρb = 7.85 ρ
28 Effect of Time Step, Relaxation and Damping t = 5e-4 Tip Displacement With Net Iteration Index t = 1e-4
29 Effect of Time Step, Relaxation and Damping
30 Flow Induced Deformation of a Beam Shin et al. (2007) b/c = 0.003 Cantilever Beam ρb/ρ = 6667 Γ = EI/ρbU2bc2 = 2 Re = 500 Plate is initially deflected in its first mode for ¼ cycle Flow Direction b = beam thickness, c = beam chord, ρb= beam density, ρ = Fluid Density, U = Free-Stream Velocity E = Modulus of Elasticity, I = moment of Inertia
31 Flow Induced Deformation of a Beam Tip Displacement with Time Vorticity Contours Explicit vs Implicit coupling
32 Plunging and Deforming Wing Heathcote et al. (2008) - Experiments Aono et al. (2009) - Computations Re = 10000 E = 200 Gpa Plunge Oscillations k = fc/u = 1.82 Deformed state ρb = 7.85 ρ (Steel Plate) Deformation along the Span-wise direction Typically 20 Inner Iterations (Correction Steps) with a t = 2.5x10-3 s
33 Plunging and Deforming Wing y y z Thrust Coefficient y x z x
34 Concluding Remarks A computational framework has been developed to couple fluid dynamics, rigid body dynamics, and structural dynamics Developed a partitioned coupling approach for fluid-structure interaction problems Importance of relaxation for partitioned coupling approaches Coupling an external structural solver with cgins under progress
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