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. 15-16 2007, North Carolina State University

SAMSI: 2007-08 Program on Random Media

Objectives and goals

• Objectives:

– 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

C

C/4

CpL.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

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

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

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 14 2186-2193 (2002)

2. He GW, M. Wang & SK Lele, PoF 16 3859-3867 (2004)

Cost Unsteady Statistics Turbulence models

DNS Unacceptable Truly representive Not necessary

LES Affordable Predictable Universal

RANS Cheap Difficult (URANS) Empirical

• Filtered Velocity: , G is a filter.dyxyGtyuV

txu ii )(),(1

),(

)(1 2

jijij

i

ij

ij

iuuuu

xu

x

p

x

uu

t

u

Large eddy simulation (LES)

velocity =large scales + small scales

computed modeled

The filtered Navier-Stokes equation

• Key issues in LES: the filtered N-S equation

1. Subgrid scale modeling: energy dissipations filter sizes2. Numerical algorithm: truncated errors grid sizes

3. Grid generation

P. Moin, Inter. J. Heat & Fluid flows, 23 (2002) 710-720

LES: a brief introduction

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. 2002 34:349-742. Computer capacity: A Pentium III 933MHz workstation with 1GB of memory

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

Directly imposed

two or more sets meshes

Hole cutting, interpolation in the overlapping region

Being explored

Immersed

Boundary

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

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

A hybrid LES and IB method

• 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

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

Two

LES+IB SGS models Wall models Wall models on Cartesian grids

One

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

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

• Calculation of : – Determined from the IB force in the tangent direction

22

2

1, ( ), 2

2

ijij t

i jij ij ijt s

j i

q S

u uC S S S S S

x x

1 exp( ),25s

yuyf y

u

y

IllustrationIllustration of IB force of IB force

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

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 (th

e viscosity terms in stream-wise and cross-wise directions)

• Poisson solver

– Pre-conditioned conjugate gradient solver

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

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

Domain size: 30Dx10Dx4D

Boundary condition:

in-flow: a uniform velocity profile

out-flow: a convective boundary condition

normal: shear free

span-wise: periodic

1.0

5.0

Shear free

Convective B

C

u=1,v=

0,w=

0

25.0

10.0

30.0

Shear free

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

Validation (I): flow around a stationary cylinder

Vortex contour

Cd Cl St

Present 1.445 ±0.342 0.169

Reference 1.35 ±0.339 0.165

Time history of drag and lift coefficients

for the flow past an rotating cylinder

Re=100Stationary

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=200Angular velocity=

Time history of drag and lift coefficients

for the flow past an rotating cylinder Vortex shedding behind an rotating cylinder

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

64 64 64

Mean velocity profile

R.m.s. velocity fluctuations

• Computation domain:

• Grid size:

• Reynolds number:

in x, y, z.

based on the wall shear velocity and the channel half-width

• 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

in x, y, z.

The IB interface is located at y=0.02

Re / 180u h v

Simulation parameters

7.0 2.0 3.5

M. Uhlmann, An immersed boundary method with direct forcing for the simulation of

Particulate flows, JCP, 209 (2005) 448-476

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:

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.

Case1: stationary

Lift and drag coefficients for the SD7003 airfoil at Re=60,000

• 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

Case1: stationary

• The dominant frequency range is from 0.2 to 3

Frequency spectra of the drag coefficient

Frequency spectra of the lift coefficient

Vortex contour

Vortex contour of a stationary airfoil SD7003, attack angle=

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 2007-4233Dye injection side views for trailing and the near-wake

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

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

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

Time history of drag and lift coefficients

Frequency spectra of the drag coefficient

Frequency spectra of the lift coefficient

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

Time history of drag and lift coefficients

Frequency spectra of the drag coefficient

Frequency spectra of the lift coefficient

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

Time history of drag and lift coefficients

Frequency spectra of the drag coefficient

Frequency spectra of the lift coefficient

Case 4: combined pitching and plunging

Vortex contour of a combined pitching and plunging airfoil SD7003

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