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**SAMSI: 2007-08 Program on Random Media**

A hybrid approach of large-eddy simulation and immersed boundary method for flapping wings at moderate Reynolds numbers Guo-wei He Department of Aerospace Engineering, Iowa State University And Institute of Mechanics Chinese Academy of Sciences Interface Problems Workshop Nov , North Carolina State University

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**Objectives and goals Objectives: Goals:**

Develop a hybrid approach of LES and IB to simulate plunge and /or pitching motions of an SD7003 airfoil at Re=60k Investigate the flow field and the aerodynamics performance of an SD7003 airfoil in plunge and/or pitching motions Goals: Develop a computational tool to predict the aerodynamics of flapping wing with experimental validation Provide quantitative documentation of the flow field and the aerodynamics performance of flapping wings by computations LES=Large-eddy simulation, IB=immersed boundary

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**A SD7003 airfoil in flapping motion**

A low speed airfoil with 8.5% thickness and 1.4% camber High frequency pitch and/or plunge motion C C/4 Cp L.E. T.E. Free stream C : chord length; L.E. : leading edge; T.E. : trailing edge; Cp : center of pitching schematical illustration of a SD7003 airfoil

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**Laminar-turbulent transition over an SD7003 airfoil**

Fixed wings: turbulent transition with separation & reattachment - The 1st stage: receptivity; - 2: Linear growth stage; - 3. Nonlinear instabilities stage; - 4. Turbulence transition stage Flapping wings: compound with turbulence - Weis-Fogh’s clap and fling; - Leading-edge vortices; - Pitching-up rotation; - Wake-capture; The challenges in numerical simulations: - Laminar-turbulent transition: turbulent flow and its transition - Plunge and/or pitching airfoil: moving boundary

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**Large-eddy simulation (LES) for turbulent & transitional flows**

LES vs DNS and RANS Time accurate LES in statistics LES correctly predicts energy spectra Subgrid scale models are developed on energy budget equation LES is being developed to predict frequency spectra or time correlations. That is a new challenge. 1. He GW, R. Rubinstein & LP Wang, PoF (2002) 2. He GW, M. Wang & SK Lele, PoF (2004) Cost Unsteady Statistics Turbulence models DNS Unacceptable Truly representive Not necessary LES Affordable Predictable Universal RANS Cheap Difficult (URANS) Empirical

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**LES: a brief introduction**

Large eddy simulation (LES) velocity =large scales + small scales computed modeled Filtered Velocity: , G is a filter. The filtered Navier-Stokes equation Key issues in LES: the filtered N-S equation Subgrid scale modeling: energy dissipations filter sizes Numerical algorithm: truncated errors grid sizes Grid generation P. Moin, Inter. J. Heat & Fluid flows, 23 (2002)

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**LES of an SD 7003 airfoil from Re=10, 000 and 1,000,000**

The number of grid points exceeds the present computer capacity; Most of the points are used to resolve inner layers Wall models needed Flapping wings: moving boundaries Grid embedding or multi-domain strategies : increase cost Unstructured grids: negative impact on stability and convergence Classic deformation or re-meshing strategies: additional overhead 1. U Piomelli & E. Balaras: Ann. Rev. Fluid Mech :349-74 2. Computer capacity: A Pentium III 933MHz workstation with 1GB of memory

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**Numerical methods for moving boundaries**

Boundary Conditions Mesh Complex/moving boundary Computing turbulence Body-fitted method Directly imposed Body-fitted Mesh-smoothing Or re-meshing Less efficient Overset method two or more sets meshes Hole cutting, interpolation in the overlapping region Being explored Immersed Method Forcing in body vicinity Cartesian mesh no more difficulty Efficiency, better conservation property Four different IB strategies for complex geometries A direction forcing at Lagrangian points Interpolation based on volume of fraction Explicit linear interpolation Ghost cell approach

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**A direct forcing approach (a IB method)**

Virtual forces are prescribed on the Cartesian grids to avoid body-fitting grids - represent the effects of body on flows - obtained to impose boundary conditions on body/flow interface IB for turbulence requires the near wall resolution in all 3 directions - local refinement Four essential steps: - track the locations of body/face interface in a Lagrangian fashion - formulate an adequate virtual force at the interface locations - transfer that force smoothly to the Eulerian grid nodes - time advancement of the Navier-Stokes equations in the Cartesian grids

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**A hybrid LES and IB method**

Bulk flows Near-wall turbulence Complex boundary geometries Challenges LES SGS models Wall models Body-fitting grids Two IB Local refinement (near wall resolution) Cartesian grids “NS + forcing” is Wall modeling LES+IB Wall models on Cartesian grids One LES+IB: LES on the Cartesian grids for complex geometries Challenges: wall modeling on the Cartesian grids - body-fitting: wall modeling in the wall normal direction - Cartesian gird: wall modeling in all three directions

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**The planned work: wall modeling**

A SGS models with a damping function - Damping functions - Eddy viscosity model Boundary layer equations - wall stress - dynamic models Shear-dependent SGS models - homogeneous shear flows - wall turbulence

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**Wall treatments in IB/LES method**

SGS model: Smagorinsky model The wall damping function is defined as: Calculation of : Minimal distance between Euler grid and Lagrangian point Determined from the IB force in the tangent direction Illustration of IB force

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**The direct forcing method**

Solve the NS equation without forcing for intermediate velocity Interpolate for Lagrangian velocity Impose boundary conditions to NS for Lagrangian velocity Interpolate the force to Eulerian grids Solve the NS equation with force for Eulerian velocity rsh = pressure + SGS residual stress + viscosity term + convection term

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**The Navier-Stokes solver**

Spatial discretization Second order finite volume method Temporal discretization Fractional step method Third order Runge-Kutta scheme is used for terms treated explicitly (the convective term and viscosity terms in span-wise direction) Second order Crank-Nicholson is used for terms treated implicitly (the viscosity terms in stream-wise and cross-wise directions) Poisson solver Pre-conditioned conjugate gradient solver

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**Validation: The 3D flow around circle cylinders**

Body-fitting grids v.s. Cartesian grids Lift and drag coefficients: vorticity behind cylinders Turbulent channel flows Benchmark problem Mean velocity profile and r.m.s velocity fluctuation

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**Validation (I and II): Flow around a circular cylinder**

1.0 5.0 Shear free Convective BC u=1,v=0,w=0 25.0 10.0 30.0 Domain size: 30Dx10Dx4D Boundary condition: in-flow: a uniform velocity profile out-flow: a convective boundary condition normal: shear free span-wise: periodic A slice of 3-D Cartesian mesh in z direction Stream and normal directions: the grids stretched to cluster points near surface Span-wise direction: uniform grids

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**Validation (I): flow around a stationary cylinder**

Cd Cl St Present 1.445 ±0.342 0.169 Reference 1.35 ±0.339 0.165 Re=100 Stationary Vortex contour Time history of drag and lift coefficients for the flow past an rotating cylinder

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**Validation (II): flow around a rotating cylinder**

Cd Cl St Present 0.809 -4.16 0.192 Overset 0.767 -3.982 0.185 Reference 0.837 -4.114 0.191 Re=200 Angular velocity= Time history of drag and lift coefficients for the flow past an rotating cylinder Vortex shedding behind an rotating cylinder

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**Validation (III): turbulent channel flow using LES and IBM**

Simulation parameters Computation domain: in x, y, z. The IB interface is located at y=0.02 Grid size: in x, y, z. Reynolds number: based on the wall shear velocity and the channel half-width Mean velocity profile Boundary condition: y=0.0,non-slip; y=h, non-slip periodic in x and z directions. SGS model: dynamic Smagorinsky SGS model IB method: direct forcing method M. Uhlmann, An immersed boundary method with direct forcing for the simulation of Particulate flows, JCP, 209 (2005) R.m.s. velocity fluctuations

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**Simulation parameters for SD 7003 airfoil**

The Reynolds number based on inflow velocity and chord length is : 60,000 Boundary conditions: inflow: uniform velocity; outflow: convective boundary condition; cross wise: shear free; span wise: periodic Four cases are simulated in present work : Case1: flow past stationary airfoil SD7003, attack angle= Case2: flow past plunging airfoil SD7003 Case3: flow past combined pitching and plunging airfoil Case4: flow past pitching airfoil SD7003 Flapping motion:

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**Simulation parameters: grid setting**

Two settings of grids are used in present case. Setting 1 . Domain size: 60C*60C*0.02C The center of the airfoil is located at (30C, 30C) Grid number: 472*332*4 Mesh size: in the uniform region (IB region): 0.005, the increase proportion is 5% in stream-wise and is 10% in cross-wise, and is uniform in span-wise. Setting 2. Domain size: 15C*10C*0.02C The center of the airfoil is located at (5C, 5C) Grid number: 570*384*4 Mesh size: in the uniform region (IB region): 0.005, the increase proportion is 2% in stream-wise,is 4% in cross-wise, and is uniform in span-wise. Setting 1 is used for all the four cases. Setting 2 is used for case 2 and case 4.

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**Case1: stationary Lift and drag coefficients consistentl with other**

author's Q3D results The transition point is 0.37C from the leading edge compared with 0.49C (W. Yuan, AIAA, 2005), due to the poor resolution near the leading edge Streamlines and turbulent shear stress for the SD7003 airfoil at Re=60,000, attack angle Lift and drag coefficients for the SD7003 airfoil at Re=60,000

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**Case1: stationary The dominant frequency range is from 0.2 to 3**

Frequency spectra of the drag coefficient Vortex contour Vortex contour of a stationary airfoil SD7003, attack angle= Frequency spectra of the lift coefficient

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**Case 2: plunging (frequency=1.25, amplitude=0.05C)**

The wake vortex structure shows consistant with experiment’s results. Present case, vortex contour behind trailing edge Expt. From Michael V.OL, AIAA Dye injection side views for trailing and the near-wake

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**Case 2: plunging (frequency=1.25, amplitude=0.05C)**

Vortex contour of a plunging airfoil SD7003, frequency=1.25, amplitude=0.05C

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**Case 2: plunging (frequency=1, amplitude=0.1C)**

The mean Drag Coefficient CD=-0.105, thrust is generated by plunging; The mean Lift Coefficient CL=0.828 For drag coefficient, the dominant frequency is f =1,2 For lift coefficient, the dominant frequency is f =1 Frequency spectra of the drag coefficient Time history of drag and lift coefficients Frequency spectra of the lift coefficient

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**Case 3: Pitching The mean drag coefficient CD=0.941**

The mean lift coefficient CL=1.235 For drag coefficient, the dominant frequency is f=2,4 For lift coefficient, the dominant frequency is f=2 Frequency spectra of the drag coefficient Time history of drag and lift coefficients Frequency spectra of the lift coefficient

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**Case 4: combined pitching and plunging motion**

The mean drag coefficient CD=0.564 The mean lift coefficient CL=1.248 For drag coefficient, the dominant frequency is f =1,2,3,4,5 For lift coefficient, the dominant frequency is f =1,2 Frequency spectra of the drag coefficient Time history of drag and lift coefficients Frequency spectra of the lift coefficient

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**Case 4: combined pitching and plunging**

Vortex contour of a combined pitching and plunging airfoil SD7003

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Summary A hybrid approach of large-eddy simulation and immersed boundary method is developed Preliminary results for a SD 7003 airfoil at Re=60,000 show the promising of the hybrid approach Wall models coupled with immersed boundary method need to be developed

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