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BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV
A NUMERICAL STUDY OF 2-D FLOW
AROUND A FLAPPING AIRFOIL
A. DUMITRACHE*
A. CARABINEANU**
Abstract: The paper explores two fundamental challenges of flapping flight.
The first part presents a computational study of the fluid physics of a heaving
airfoil to elucidate the important features of these flows and determine how
the flow physics relate to important parameters such as power consumption
and efficiency. The second part presents a method for creating a control
model for flapping flight. Using proper orthogonal decomposition (POD),
sets of orthogonal basis functions are generated for simulating flows at the
various heaving and pitching parameters. The effects of these approximations
on the accuracy of the model are assessed.
Keywords: Flapping airfoil, Navier-Stokes equations, Proper orthogonal
decomposition.
1. Introduction
The oscillation of an airfoil-like appendage —"flapping" — is a common form of
propulsion for many flying animals or many swimming species.
The generation of thrust by an oscillating airfoil has a notable effect on the qualitative
structure of the wake. Thrust production by an oscillating airfoil was first described in the
early part of the 20th century. [Knoller, 1909] and [Betz, 1912] independently noted that
the vertical motion of an airfoil produces an effective angle of attack so that the resulting
normal force vector has a component in the forward direction.
[Lai and Platzer, 1999] and [Jones et al., 1996] used a water tunnel and laser Doppler
velocimetry to obtain high-resolution velocity measurements in the wake of a purely
heaving airfoil. Their experimental results confirmed the calculations of [Triantafyllou et
at., 1993], which identified the Strouhal number, St  fA U   as an important parameter
for thrust generation. Here, f is the oscillation frequency, U  the free stream velocity,
and A is twice the amplitude of motion measured from the mean position. [Lai and
Platzer, 1999] noted that the structure of the wake (drag or thrust) can be classified
according to the Strouhal number, with those motions below a threshold of St  0.06
producing drag and those above producing thrust.
[Jones et al., 1998] developed an inviscid model to determine the wake structure on a
two-dimensional airfoil undergoing pitching, heaving, and combined motions. More
recently, [Minotti, 2002] developed a model of a flapping flat plate from two-dimensional
potential theory that included a stationary vortex near the leading edge to maintain a
regular velocity at the singularity at the leading edge.
None of these studies fully address the importance of viscous effects. Of course, as the
length scales of flapping flight shrink, viscosity will play a greater role in the fluid
*
Institute of Mathematical Statistics and Applied Mathematics “Gheorghe Mihoc-Caius Iacob”.
**
University of Bucharest, Department of Mathematics
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126
mechanics.
2. Problem definition
It is considered a rigid, two-dimensional airfoil undergoing prescribed, periodic,
flapping motions at zero angle of attack in an incompressible, viscous flow with constant
horizontal (chord-wise) free stream velocity, U  . The flapping consists of heaving
(vertical), lagging (horizontal), and pitching (angular) motions and the airfoil is in cruise
conditions: level flight with constant average velocity, U  .
Fig. 2.1 Newtonian and airfoil coordinate systems
For computational simplicity, the problem is formulated in a non-inertial reference
frame fixed to the airfoil such that the origin coincides with the pitch-axis, the x-axis is
fixed parallel to the chord, and the y-axis is perpendicular to both the x-axis and the span
(see Figure 2.1). In the inertial frame, the X-axis is horizontal (mean flow direction) and
the Y-axis is vertical. The dimensional horizontal, vertical, and angular positions of the
airfoil are denoted by X 0 , Y0 , and  , respectively, where the first two are made nondimensional by dividing the absolute position, X 0 , Y0 , by the chord length, c.
2.1. Kinematics
Introducing the reduced frequency, k  2 f c / U  , the position of the airfoil for
sinusoidal motions is given as
Y0  h sin k,
X 0  l sin k  ,
  max sin k   ,
(2.1)
where h  Ymax c is the dimensionless heave amplitude, l  X max c is the dimensionless lag
amplitude, and   U  t c is the dimensionless time,  and  are dimensionless phase
parameters for lagging and pitching, respectively. By differentiating Equation 2.1, the
dimensionless heave velocity can be expressed as V0  Y0   kh cos k . Note that the
maxi-mum heave velocity is given by the quantity kh= 2fYmax / U  , which differs from
the definition of St by a factor of  . Being more intuitive, the quantity kh is preferred for
categorizing the results, although conversion to St will be made when appropriate
Bulletin of Transilvania University of Brasov Vol 13(48) - 2006
127
2.2 . Governing Equations
In the non-inertial frame, the non-dimensional Navier-Stokes equations are written:
u
1 2
 p  u   u 
 u  a xyz t 
t
Re
(2.2)
where a xyz is the acceleration of the non-inertial reference frame (including linear,
centripetal, rotational and Coriolis terms).
Taking the curl of Equation 2.2 and defining the vorticity in two-dimensions as
    u   kˆ (where k̂ is the unit vector perpendicular to the plane of flow), results in
the vorticity transport equation:

1 2
   u  
   2
t
Re
(2.3)
Note that the only source of vorticity from non-inertial effects is from the angular
acceleration of non-inertial reference frame, denoted by  . This observation is used to
simplify matters by defining vorticity to be the sum of a background vorticity, Z, and a
disturbance term, ' :   Z  ' where:
Z  2
(2.4)
Substituting these into Equation 2.3 results in:
'
1 2
 u  ' 
 '
t
Re
(2.5)
where a simplification is made from the fact that the gradient and Laplacian of the
spatially-constant quantity Z are zero.
Assuming incompressibility, Equation 2.2 is supplemented by the continuity
equation   u  0 and after defining the stream function,  , by
u     kˆ
(2.6)
it can be related to  through the continuity equation
2  
(2.7)
As with vorticity, the stream function and velocity components are decomposed into
combinations of background and disturbance terms. The most convenient choice is to use
the free stream velocity, U  , as the background value, so that
u  U   u'
(2.8)
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and     ' such that
 
U     kˆ ;
 
u '     ' kˆ
(2.9)
For a constant free stream velocity, U  , the equation for  is given by

1
  U   U 0 x sin   y cos    V  V0  y sin   x cos     x 2  y 2
2

(2.10)
plus an arbitrary constant taken to be zero. Note that  2   2   Z and the stream
function equation can be reduced to
2'  2  2     Z  '
(2.11)
That is, the problem has been reduced to two analytical expressions for the background
values (Equations 2.4 and 2.10) and two partial differential equations: a vorticity
transport equation for  ' (Equation 2.5) and a Poisson equation for  ' (Equation 2.11).
The interaction between these two sets of equations is in the advection term for  ' ,
which requires the combined background and disturbance velocity (Equation 2.8).
2.3 Boundary Conditions
The airfoil is a no-slip surface so that the total vorticity at the airfoil,  a can be related to
the total stream function by
a  
 2
n 2 airfoil
(2.12)
where  refers to the normal derivative. Thus,
n
 'a  
 2 '
 2

Z
2
n airfoil
n 2 airfoil
(2.13)
For the infinite-space problem, the appropriate boundary conditions at infinity are that the
velocity equals the free stream velocity and the fluid is irrotational in the inertial frame.
Thus,
'   0,
  ' kˆ  0.
(2.14)
3 Numerical Implementation
3.1 Discretization of the Navier-Stokes Equations
The governing equations are discretized using finite differences on a conformal map. A
rectangular  ,  domain is first mapped to a circular domain using a log-polar
transformation, and the circular domain is mapped to an airfoil by means of a Joukowski
transformation.
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129
Before discretization, the governing equations are transformed from physical x, y
space to computational  ,  space. The vorticity transport equation becomes
 '
1 2
h1 h2
   ,     ,  '
  ,  ' ,
(3.1)
t
Re
where the subscript   , refers to derivatives in the  ,  domain, h1 and h1 are the grid
transformation metrics which relate the size of a grid cell in the x, y domain to the  , 
domain.
Additionally, the Poisson equation for  ' becomes
2 , '  h1h2 '.
(3.2)
Stepping in time is performed using a second-order Runge-Kutta scheme (see [Schneider
and Zedan, 1981]).
3.2 Boundary Conditions
Boundary conditions are required for  ' (Equation 2.3) and  ' (Equation 2.11). At the
airfoil. Equation 2.13 is discretized using a second order difference equation:
 'a 
 2  8 1  7 0 
2n 2
Z
At the inlet, the flow is assumed to be irrotational and so  ' 0 . At the outlet, viscous
effects are neglected so that the material derivative of the vorticity vanishes.
For the stream function,  '   on the airfoil. At the outer boundary, the normal
derivative of  ' is prescribed, which amounts to specifying the tangential velocity. At the
inlet, disturbances to the free stream flow are neglected so that
 '
0
n inlet
(3.3)
If viscosity is again neglected, then the material derivative of velocity is affected only by
the pressure gradient. Thus, at the outlet
D   ' 
 0,


Dt  n  outlet
(3.4)
which has a similar form to the boundary condition for vorticity.
3.3 Forces and Power
The total force on the airfoil is a combination of pressure and viscous forces. A
simplified procedure for calculating the pressure on the airfoil is based on the relationship
p
1 

 a  ds
(3.5)
s
Re n
which relates the pressure gradient along a no-slip wall to the normal derivative of
vorticity, modified for the d'Alembert forces.
The viscous force is found from the shear stress on the airfoil,  s which is related to the
A numerical study of 2-d flow around a flapping airfoil
130
tangential velocity, u s by
n 
1 u e
1
  a
Re n
Re
(3.6)
Once the pressure and viscous forces are known at each point, they are integrated
numerically to find the force and moment on the airfoil, which can be easily resolved into
the force components, FX and FY, and moment, M, in the inertial reference frame.
For the calculation of average forces, it is convenient to have a whole number of steps
in each flap, which has the duration   2 / k . The time step,  is then   2 /( nk ) ,
where n is the number of steps per flap.
The instantaneous power can be calculated from the total force by multiplying the force
components by the appropriate velocity components in the inertial frame. The input
power coefficient, Pi is the combination of the power needed to heave, lag, and pitch the
airfoil:
(3.7)
Pi  FY V0  FX U 0  M
Note that the power input is negative by this definition. The output power coefficient,
Po , from thrust is
Po   FXU 
(3.8)
so that the output power is positive when thrust is produced and negative for drag.
Integrating the instantaneous power yields the total work required (input) to flap the
airfoil, Wi , and the work done in propelling it (output), W0. Dividing the work by the time
over which the power is integrated (which can range from one stroke to several flaps),
yields the average power. Following [Jones and Platzer, 1997] and [Wang, 2000], the
thrust efficiency,  , is defined as the ratio of total work out to total work in:    Wo Wi .
3.4 Validation of the Code
The performance of the model was assessed through several test simulations.
Preliminarily, flow around a stationary circular cylinder was simulated at Re  100 using a
128  384 grid. The simulated St was 0.164 and the force coefficients on the cylinder were
found to be CD, avg  1.31 , (average drag coefficient), CL, rms  0.226 (rms lift coefficient) and
C L ,max  0.319 (maximum lift coefficient)
To assess the dynamics of the vortex wake, the simulated flow behind an impulsively
started cylinder was compared to the experimental results of [Bouard and Coutanceau,
1980] Re  550 .
Additionally, the computational and experimental velocities along the mean wake line
were compared for the same flow conditions as above.
Bulletin of Transilvania University of Brasov Vol 13(48) - 2006
131
Fig. 3.1. Computed vorticity at the leading edge (  LE ) and trailing edge (  TE ) for
various grid resolution
To ascertain the effects of grid refinement on the solution, a heaving airfoil was
simulated with kh  4.0 , h  0.25 , and Re  1000 using a number of grids. Figure 3.1
shows the vorticity at the leading and trailing edges after one stroke for various levels of
grid refinement (where the relative grid spacing of 1 corresponds to a 512  512 grid).
4 Results of the CFD Model
The numerical model described in the previous section was used to simulate a heaving
airfoil with frequencies ranging from 2.0  k  10.0 and maxi-mum heave velocities
from 0.8  kh  1.5 0.25  St  0.48  at Re = 500. A wide variety of flow patterns are
observed in the parameter space studied. The long-term solutions for flows with kh  0.8
are periodic and symmetric, meaning each up-stroke and down-stroke are mirror images.
For kh  1.0 , asymmetric solutions are observed. For kh  1.2 , aperiodic, quasiperiodic, and asymmetric solutions are common.
The term aperiodic is self-explanatory; by quasi-periodic it is meant that for several
periods the flow changes relatively little from period to period until there is a large
qualitative change lasting for a few strokes, after which the flow returns to its former,
slowly evolving state. Asymmetric solutions are those where there is no symmetry in
either the forces on the airfoil or the vorticity in the wake in the direction of heaving, as
when the wake is deflected from the mean heave position.
4.1 Flow Patterns
During the first half of the stroke, vortex filaments form at both the leading and trailing
edges. At the leading edge, the vortex filament begins to roll up into a vortex at midstroke. As the airfoil motion reverses, the vortex filament weakens and a significant
secondary vortex extends from the airfoil, cutting off the filament feeding the leading
edge vortex (LEV). The LEV then advects downstream.
As the heaving frequency increase, the time scale of the heaving motion approaches the
hydrodynamic time scale flow. As the frequency increases further, the LEV remains near
the leading edge for a larger portion of the subsequent stroke where its strength is reduced
substantially by interaction with the airfoil.
The asymmetry in the flow leads to an asymmetry in the power coefficients, as well.
A numerical study of 2-d flow around a flapping airfoil
132
At high frequency, the flow are highly aperiodic with large wake deflections.
Analysis of the flow shows that the high frequency of oscillation allows successive
trailing edge vortices to have significant interaction with one another.
4.2 Power and Efficiency
Because many of the simulations produce aperiodic solutions, averages are taken over
many flaps (starting after several flaps are made to allow for the suppression of the initial
transients). The overall efficiency is very low: max  11% at k  5.333 and kh  1.2 . In
comparison, [Wang, 2000] found the maximum efficiency for a thin ellipse to be
 10% at similar values of frequency and amplitude.
Figure 4.1 shows the horizontal (output) power components for two different
simulations with kh  1.0 , k  2.0 (low efficiency) and k  5.333 (high efficiency).
The horizontal axes are scaled by the appropriate k so that the strokes from each
simulation coincide. For the k  5.333 case, the output power rises and falls smoothly
and is roughly correlated with the velocity of the airfoil; peak power occurs when the
airfoil is near the mid-stroke and is lowest when the airfoil is near the extremes. For the
k  2.0 case, the output power initially rises at the start of each stroke but before the
airfoil reaches mid-stroke, the power temporarily levels off, after which it remains
substantially below the k  5.333 case.
Interaction between the airfoil and the shed vortices induces a drag on the airfoil, which
has a stronger effect at higher frequencies due to the greater proximity of the vortices.
There is a strong correlation between heaving efficiency and the separation between the
heaving frequency for a given profile and the frequency resulting in the maximum spatial
amplification of this profile. As the heaving frequency increases, the driving frequency
becomes greater than the frequency of the most unstable mode.
Fig. 4.1. Output power for k = 2 and k = 5.333
over one heaving cycle.
5 Analysis of the CFD Model
Over the parameter ranges covered in this numerical study, several distinct solution
topologies are observed, including aperiodic and asymmetric solutions. In their
comprehensive experimental investigation of flow around an oscillating cylinder,
[Williamson and Roshko (1988)] identified a number of different flow regimes similar to
those observed in this work, including deflected wakes as well as various numbers of
vortices shed into the wake per oscillation, symmetrically or asymmetrically.
Bulletin of Transilvania University of Brasov Vol 13(48) - 2006
133
Additionally, for a large portion of their parameter space, they found no stable pattern.
Whereas [Williamson and Roshko, 1988] found sharp changes between the various
flow regimes in their study of a circular cylinder, the results here demonstrate that for a
heaving airfoil the transition regions are often characterized by aperiodic flows that
"switch" between the neighboring flow regimes. Additionally, at high kh and high k, all of
the simulations produced aperiodic results.
The fate of the LEV can also be correlated to the heaving efficiency. While the overall
efficiency was low, large increases in efficiency occur at the transition from a shed LEV
to one that is dissipated. The results are also in accordance with [Gustafson, 1996] in that
some of the vorticity contained in the LEV is subsequently recaptured by the airfoil.
In agreement with [Anderson et al., 1998], high thrust coefficients and propulsion
efficiencies corres-pond to the positive reinforcement of the trailing edge vortex (TEV)
by the LEV. Additionally, thrust and efficiency are greatly reduced when there is negative
reinforcement between the LEV and TEV. [Tuncer and Platzer, 1996] noted a large
increase in efficiency is possible when a stationary airfoil is placed in the wake of a
heaving airfoil. As the heave frequency is increased, the wavelength of the wake vortices
is shortened and shed vortices remain nearer to the trailing edge for a greater portion of
each flap, and the increased interaction leads to lower efficiency [Jones and Platzer,
1997].
While the current study focuses on heaving motions, it is expected that analogous
relationships will hold for motions with pitching and lagging. With pitching, the behavior
and evolution of the LEV can be controlled more effectively. Absent that in twodimensions, the pitching of the wing will assist in the streamlining of the motion and
control of the LEV.
6 A Reduced-Order Model
In the realm of flapping flight, recent experimental studies by [Srygley and Thomas,
2002] and [Fry et al., 2003] have shown that the forces generated by flapping insects are
very sensitive to the wing kinematics. Since active control requires a solver that can run
at the time-scale of the fluid motion, Navier-Stokes (N-S) solvers are not suited for active
control of fluids. Alternatively, reduced-order models can be used to greatly simplify a
fluid problem and drastically cut the computational resources needed to solve it.
Reduced-order models, however, result in the loss of some information about the flow,
thus adding another level of approximation to the problem.
The focus of this second part is the development of a reduced-order model that can
potentially be used in the active control of a flapping wing.
One method that has shown promise uses proper orthogonal decomposition (POD,
described in the next section) to generate a set of orthogonal modes from a given flow
(either experimental of simulated). These modes can then be used in a Galerkin projection
of the N-S equations. The POD basis functions are optimal in the sense that subsequent
modes contain the maximum amount of remaining energy from the data set. In this way,
it is possible to represent a complex flow using only a few basis functions, as opposed to
the thousands of mesh points needed to determine a flow using traditional computational
techniques. The result is to convert the non-linear partial differential N-S equations to a
(comparatively small) set of coupled, non-linear, ordinary differential equations (ODE's).
A typical CFD simulation of a flapping airfoil presented in this paper requires on the
order of 30,000 degrees of freedom. The same simulation can be closely approximated by
as few as ten to twenty ODE's.
Recently, several authors have attempted to demonstrate the usefulness of POD for
A numerical study of 2-d flow around a flapping airfoil
134
generating control models.
7 Derivation of the Reduced-Order Model
7.1 Proper Orthogonal Decomposition
Proper orthogonal decomposition [POD], also known as Karhunen-Loeve expansion,
was introduced in the context of turbulence by Lumley, where it was used to identify and
examine the stability of coherent structures in turbulent flows (see [Holmes et al., 1996]
for a review). With POD, a set of modes is produced from a series of flow snapshots
generated by either computational or (less often) experimental means. Either the vorticity
field,  or the velocity field, u, can be used for the POD. When velocity is used, the
eigenvalues of the decomposition are a direct measure of the kinetic energy contained in
each mode and the POD basis functions are optimal in the sense that they capture the
most kinetic energy possible for a given number of modes. The velocity formulation
proved superior to the vorticity formulation for the current work, and only the former
derivation is provided here.
Let u k  x  represent a snapshot of the velocity field over domain D at time k and
ui , u j
be a suitable defined inner product. Following [Ravindran, 2000], a function

is sought that gives the best representation of the ensemble of N snapshots
k
L  u : 1  i  N in that it maximizes
L2

1
N

ui , 
N

, 
i 1
2
(7.1)
.
In other words, one seeks a function that has the largest mean square projection of the set
L . To cast the problem into an equivalent eigenvalue, define
1
N
K 
so that
 D u i x  u i x'x' dx'
N
(7.2)
i 1
K,  
N

1
N
2
ui , 
. Also define
i 1
K, 
, 

1
N
N

2
ui , 
i 1
, 
 .
Let  * be a function that maximizes  . Writing variations of  * as *  ' ,  can be
written as
F   
dF d   0 and so
K * ,  *   K * ,  '   K ' ,  *   2 K ' ,  '
* , *   * ,  '    ' , *   2  ' ,  '
K * ,  '    * ,  '
. The maximum occurs where
.
Since  ' is an arbitrary variation, the maximization problem in Equation 7.1 is the same
as solving the eigenvalue problem
(7.3)
K*  *
Assuming  has the form (i.e., the modes are linear combinations of the snapshots)

N
 w j u j , and using Equation 7.2, Equation 7.3 can be written
CW  W , where
j 1
Cij 
D u i  u j dxdy
and W is the matrix of eigenvectors. By construction, C is non-
negative Hermitian and the eigenvectors in W are orthogonal.
Bulletin of Transilvania University of Brasov Vol 13(48) - 2006
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For each eigenvalue of C,  j , there is a corres-ponding mode,  j . Using J modes,
ordered by  j , the snapshots can be approximated by
u k x   uˆ k x  
J
 y j k t  j
(7.4)
j 1
where û is the approximate velocity profile and the coefficients, y j , are given by
y j kt  
1
uk ,  j
j
(7.5)
Equation 7.5 will be referred to as a projection of a solution, since it is the result of
projecting a snapshot u k onto the POD modes,  j . It is convenient to normalize the
modes such that i , i  1 , making the modes orthonormal.
7.2 Galerkin Projection
The POD modes can be used to simulate a fluid flow by reformulating the N-S equations
using a Galerkin projection, in the non-inertial reference frame of the airfoil. Given the
appropriately modified Navier-Stokes equations,
u
1 2
 p  u   u 
 u  a 0 t   kˆ  r   2 r  2kˆ  u ,
t
Re
(7.6)
the time dependent velocity field, u  x, t , is expanded as
u  x, t  
M

 m t u m x  
m 1
J
 y t  x
j
j
j 1
(7.7)
where J is the number of POD modes and M is the number of control functions needed to
simulate the desired motion (e.g., heaving, pitching, or lagging; described below). Note
that  m t  are prescribed control inputs, while y j t  are the dependent variables of the
simulation. When control functions are used, the snapshots must be modified by
subtracting the control functions before the decomposition is performed:
M
k
umodified
 uk 
  m kt um .
m 1
Using [ ] to represent a boundary integral and noting that  i ,  2 u    i , u   i u ,
Equation 7.7 is projected onto the orthogonal modes,  i , to get:
u
1
1
  i ,u   u 
 i , u   i u    i , a 0    i , ˆ  r
t
Re
Re
(7.8)
2
   i , r  2  i ,  i , ˆ  u .
The pressure term does not add any terms to Equation 7.8. Because the modes are linear
combinations of the divergence free snapshots, each mode is itself divergence free.
Therefore, one can write  i , p     p i , 1  p i   0 .
The last step follows from the fact that the velocity on the airfoil is zero in the noninertial reference frame, and the pressure at the outer boundary is considered negligible.
By orthogonality, the left hand side of Equation 7.8 reduces to:
i ,
i ,
u

t
 m
y ~
 i , um  i  i ,
t
m 1 t
M

A numerical study of 2-d flow around a flapping airfoil
136
~
where  i , is retained to maintain the generality of the following derivation, even though
~
in practice the modes are normalized and  i  1 .
For the flapping airfoil, the prescribed control inputs are heaving, lagging, and angular
(pitching) velocity. Thus, to use the control notation of Equation 7.7:
 h  V y ;  l  V x ;  p   .
After much algebra, the N-S equations are reduced to the following set of ODE’s:
M
J
J

1 J
1 M  m eim M J
~ y
i i   m a im   y j bij   y i y k cijk   p2 dˆ ip 
 m y j f ijm


t m 1 t
Re j 1
Re

j 1 k 1
m 1
m 1 j 1
(7.9)
M M
M
J

   m  n g imn   p  y j hi j    m hˆim 
m 1 n 1
m 1
 j 1

where

aim   i , um ; bij    i ,u j   i u j



cijk    i , u j   u k ; d ii    i , iˆ
d ih    i , ˆj ; d ip    i , kˆ  r
(7.10)
dˆip    i , r ; eim   i ,um  i um 


f ijm    i , u j   u m   i ,u m  u j
g imn    i ,um  un
hi j  2  i , kˆ  u j ;
hˆim  2  i , kˆ  u m
where the superscripts m and n refer to any of the motions being controlled. Equations 7.9
and 7.10 constitute an initial value problem with coupled, non-linear ODE's with constant
coefficients a im , bij , etc. Solving these equations with a suitable time-stepping algorithm
will be referred to as a Galerkin simulation. The velocity field produced by solving these
equations will be denoted by u~x, t  .
7.3 Definition of Errors
Following [Graham et al., 1999], errors are classified into two categories, projection
errors and simulation errors. Projection errors result from the inability to exactly recreate
a snapshot using the POD modes of a given simulation. For this work, the projection error
is defined as:
p 
u  uˆ, u  uˆ
u
m 1  mum , u  m 1  mu m
M
M
(7.11)
where M is the number of control functions (depending on the motions - heaving,
pitching, or lagging - to be controlled).
Simulation errors result from errors in yi , generated in the Galerkin simulation. They
include discretization errors, numerical (round-off) errors, and most importantly, errors
resulting from the loss of information about the flow during the POD. The simulation
error is defined as the difference between the total error and the projection error,
Bulletin of Transilvania University of Brasov Vol 13(48) - 2006
 s   t   p , where the total error is given by
u  u~, u  u~
t 
M
M
u
 mum , u 
 mum
m 1
m 1
137
(7.12)
Ultimately, the measure of the success of the model will be its ability to reproduce the
forces and moment on the airfoil.
7.4 Control Modes
To be able to change the flapping parameters in this formulation, it is necessary to
include a control function in the POD and subsequent Galerkin simulation. While the
restrictions on the control function are fairly loose, the actual control function chosen can
have a marked effect on the accuracy of the method. The control function is required to
be divergence free and need only satisfy the appropriate boundary conditions; otherwise,
it is arbitrary.
Several methods were used to generate candidate control functions. The simplest was to
impulsively start an airfoil with the desired motion in a quiescent fluid and use an
arbitrary snapshot from the generated flow field. Additional control functions were
generated by oscillating an airfoil and subtracting off either the mean flow or the flow
around a stationary airfoil.
The excellent agreement among the mode coefficients leads to very small simulation
errors. While the errors using the control function from the impulsively started airfoil are
also low, using a heaving airfoil less the stationary flow substantially reduces the errors.
Note that the control function was generated from a simulation with heaving parameters
identical to the test case.
8 Results of the POD/Galerkin Simulations
The POD/Galerkin model described above is used to simulate a flapping airfoil at a
variety of parameters. For each of the cases presented in this section, the POD modes
were created by decomposing snapshots from one or more CFD simulations.
The first demonstration of the reduced-order approach will focus on heaving only. As
presented before, for certain heaving parameters the flow can be asymmetric or aperiodic.
The aperiodic solutions cause serious problems for the POD approach, since the number
of flaps (and thereby snapshots) required to include all the information in the flow grows
substantially (infinitely, for chaotic solutions).
Additionally, the asymmetric flows are dependent on starting conditions; while this
could make for an interesting bifurcation analysis it is not the focus of this work.
Therefore, only the symmetric, periodic heaving-only simulations will be discussed here;
chaotic flows can be "stabilized" by including a pitching component.
Among other factors, the accuracy of a Galerkin simulation is dependent on both the
number of modes retained in the simulation and the number of snapshots used to generate
these modes. As the number of snapshots is increased, the dynamic of finer flow details
can be captured and the projection error is reduced.
To investigate the effect of the number of snapshots on the error components, several
additional test simulation are made using the same flapping parameters as above. As is
evident, the simulation error typically increases when fewer snapshots are used in the
decomposition.
138
A numerical study of 2-d flow around a flapping airfoil
8.1 Variation of the Heaving Parameters
To be a useful control model, it is necessary that a Galerkin simulation be able to
represent a range of flapping parameters that may differ from those used to create the
POD modes. Thus, it is necessary that a set of POD modes be able to accurately
reproduce the forces on the airfoil for a range of parameters in some vicinity of the
parameters used to create the snapshots.
For heaving motions, three non-dimensional para-meters can be used to describe the
flow: k , h and Re , but the heaving control studies will be limited to k and h , the
primary parameters of the heaving motion.
There are a multitude of ways to construct the POD mode sets. One can simply use
POD modes generated from a single CFD simulation with certain parameters and then
alter  t  in Equation 7.9 to simulate a new set of parameters. Errors will be introduced
into the Galerkin simulation because the modes do not contain all the necessary
information about the new flow: i.e. they do not span the solution space for the new
parameters. On the other hand, by combining snapshots from two or more CFD
simulations, the information contained in the mode set can be enhanced.
Thus, it is expected that, at least for the periodic and symmetric region of the parameter
space, the sensitivity of the Galerkin simulation to changes in k and h is roughly
constant.
8.2 Pitching
As noted in Section 4, for certain sets of heaving parameters the resulting flow becomes
aperiodic and the generation of a standard mode set becomes impractical. In particular,
the parameters that produce the highest efficiency are either in or very near to the
aperiodic regions.
By allowing pitching in the airfoil motion, however, the region of periodic solutions can
be extended further into the more energetic flapping regimes.
Additional control functions were generated in a manner similar to the creation of the
control function for heaving; an airfoil was oscillated in pitch or lag in a steady free
stream, and the flow about a stationary airfoil in an identical free stream was later
subtracted to produce a control mode with the desired motion. For the pitching cases
presented here, the pitch-axis is at the leading edge and the pitching motion lags the
heaving motion by  2 .
If the simulation of an airfoil with k  5.56 , h  0.25 ,  max  0.2 , were for heaving
only, the frequency and amplitude would place it in the aperiodic region of the parameter
space. However, by adding the pitching motion, the flow is both periodic and symmetric.
The errors are in range of 10%. Additionally, the efficiency of the flapping motion is
nearly double that of the heaving only case; for the given parameters,   18% .
There are many parameters needed to define the motion of a flapping wing. Even a twodimensional wing undergoing sinusoidal motion requires specifying the Reynolds
number, frequency, the heave, lag, and pitch amplitudes, the lag and pitch phase angle,
and location of the pitch axis.
A comprehensive assessment of all of the parameters is not practical within the scope of
this work. However, it is useful to assess the ability of the POD/Galerkin technique in
simulating variations in more than one flapping parameter. To demonstrate this concept, a
Galerkin simulation is performed at k  5.56 , h  0.25 , max  0.2 using POD modes
Bulletin of Transilvania University of Brasov Vol 13(48) - 2006
139
generated from snapshots of CFD simulations with k  5.26 , h  0.25 , max  0.1875 ,
= 0.1875; and
max
k  5.26 ,
h  0.25, max  0.2125 ; k  5.88,
h  0.25,
k  5.88, h  0.25, max  0.2125 .
Note that the CFD simulations vary in both flapping frequency and pitch amplitude and
share neither of these parameters with the Galerkin simulation.
Figure 8.1 shows the coefficient errors and force portrait for this simulation using 36
modes ( M 1  104 ). For the first flap, the forces track nearly as well for this simulation
as for pitching only, demonstrating that the control of multiple parameters is as feasible as
independent parameters.
Fig. 8.1. – (a) Projection, simulation, and total errors for a Galerkin simulation with
k  5.56 , h  0.25 ,  max  0.2 using a POD mode set created from CFD simulation;
(b) Forces for the same simulation.
9 Assessment of the Reduced-Order Model
The cases presented in Section 8 demonstrate that the flow around a flapping wing and
the forces generated by the flapping motion can be modeled accurately using the
POD/Galerkin approach described in Section 7. Variation of flapping parameters can be
accommodated, but the parameter range is dependent on the method used to construct the
POD mode sets.
Section 8 demonstrates that mode sets created from individual CFD simulations are not
useful for variations in flapping parameters. By combining snapshots from multiple CFD
simulations, however, the Galerkin simulations can provide reliable results for small and
moderate variations in flapping parameters. Thus, to simulate moderate changes in
140
A numerical study of 2-d flow around a flapping airfoil
flapping parameters, two alternatives are possible: the first is to use many POD mode sets
with narrow parameter ranges and switch among the sets as necessary; the alternative is
to generate one large mode set with a wide range of parameters. The preferred method
will depend on the structure of the controller. Using more snapshots requires more
preprocessing and solving larger sets of equations. The preprocessing is unimportant in
that it only needs to be performed once. More restrictive is increasing the size of the
equation set; larger equation sets will be more computationally intensive for the
controller. For example, a 25% change in frequency is achieved using 35 modes
generated by combining snapshots from four CFD simulations. Therefore, there is a
trade-off between the size of the mode set (and corresponding numbers of equations in
the Galerkin simulation) and the useful range of that mode set.
While the overall error of the Galerkin simulations is small (the difference in energy
between the projected and simulated flows is often less than 10%), for some simulations,
the forces deviate noticeably before the completion of one flap and become totally
inaccurate after several flaps. The lack of accuracy in long-term prediction, however, is
not a great impediment to the use of the POD/Galerkin approach for a control model.
Indeed, if only long-term solutions (i.e., over several flaps such that the flow reaches a
periodic state) are needed, much more efficient, if not as elegant, solutions are possible
(most notably, a simple lookup table with flapping parameters as inputs and forces and
moments as outputs).
However, for active control at time-scales at or less than one flap, the long term
inaccuracy is not as important, given a method for observing the flow and feeding it back
into the controller; feedback loops are routinely used to account for the inability of a
model to perfectly account for the physics of the controlled system.
Observing the flow and generating feedback loops is no small task, however. The
approach described above relies on global observability. That is, the POD modes are
created from (spatially) complete know-ledge of the velocity field. In practice,
information about the flow would be limited to local properties near the airfoil such as
pressure, velocity, or shear, or integrated properties such as total forces on the airfoil.
Thus, a feedback loop could not use global knowledge of the velocity field to recalibrate
the Galerkin coefficients (e.g., by using the projection in Equation 7.5).
10 Conclusion
This paper focuses on two important aspects of flap-ping flight: the physics of the flow
of a fluid around a heaving airfoil and the development of a reduced-order model for the
control of a flapping airfoil.
A numerical study for two-dimensional flow around an airfoil undergoing prescribed
oscillatory motion in a viscous flow is developed. The model is used to examine the flow
characteristics and power coefficients of a symmetric airfoil heaving sinusoidally over a
range of frequencies and amplitudes. Both periodic and aperiodic solutions are found.
Additionally, some flows are asymmetric in that the up-stroke is not a mirror image of the
down-stroke. For a given Strouhal number, the maximum efficiency occurs at an
intermediate heaving amplitude. Above the opti-mum frequency, the efficiency decreases
similarly to inviscid theory.
The computational model is used as the basis for developing a reduced-order model for
active control of flapping wing. Using POD, sets of orthogonal basis functions are
generated for simulating flows at the various heaving and pitching parameters. With
POD, most of the energy in the flow is concentrated in just a few basis functions. Even
two-dimensional flapping requires a large number of parameters, only a few of which are
Bulletin of Transilvania University of Brasov Vol 13(48) - 2006
141
examined here. The results of Section 8 demonstrate that variations of two parameters ( k
and max ) can be modeled as well as variations of single parameters when the snapshots
are taken from CFD simulations that vary in both parameters. Incorporating all of the
parameters in two-dimensional flapping into a comprehensive set of POD modes would
require a substantial amount of preprocessing, as well as large sets of modes to span all of
the control space. Fixing certain parameters in the hardware would alleviate this problem
to some extent.
The suitability of this approach for controlling a flapping wing over a broad range of
parameters is analysed.
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Un studiu numeric al curgerii 2-D in jurul unui profil batant
Rezumat Articolul exploreaza doua caracteristici fundamentale ale zborului
cu profil batant (zborul pasarilor). Prima parte prezinta un studiu
computational al fizicii fluidelor unui profil cu miscare de “bataie” pentru a
elucida caracteristicile importante ale acestor curgeri si a determina cum este
raportata fizica curgerii la parametri importanti, cum sunt consumul de putere
si randamentul. A doua parte prezinta o metoda de obtinere a unui model de
control pentru acest tip de zbor. Folosind metoda de descompunere
corespunzatoare (POD) , sunt generate un set de functii de baza ortogonale
pentru simularea curgerii pentru diferiti parametrii de bataie si tangaj a
profilului. Sunt determinate
efectele acestor aproximatii asupra
preciziei modelului.
Cuvinte cheie: profil batant, ecuatii Navier-Stokes, descompunere
ortogonala (POD)