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