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Thepvongs-AHS_Forum2009_GIT.pdf
Computational Aeroelasticity of Rotating Wings with Deformable
Airfoils
Smith Thepvongs
James R. Cook
Graduate Research Assistant
Graduate Research Assistant
[email protected]
[email protected]
Carlos E.S. Cesnik
Marilyn J. Smith
Professor
Associate Professor
[email protected]
[email protected]
Department of Aerospace Engineering
School of Aerospace Engineering
University of Michigan, Ann Arbor, MI Georgia Institute of Technology, Atlanta, GA
Abstract
This paper presents a simulation for high-fidelity aeroelastic analysis of rotating wings with camber-wise
structural flexibility and embedded actuators. An unstructured Reynolds-Averaged Navier-Stokes (RANS)
computational fluid dynamics (CFD) solver is coupled with a non-linear structural dynamics analysis. The
CFD solution uses overset grids to combine the stationary and moving frames of reference. The structural formulation expands the conventional one-dimensional beam representation with additional degrees-of-freedom
to capture plate-like cross-sectional deformations while allowing an arbitrary distribution of active and passive materials in the cross section. Motion and forces on the non-coincident fluid and structural grids are
transferred using a finite-element-based interpolation, along with a least-squares fit for extrapolations. Trim
and convergence to periodic response are assisted by a low-order analysis that is also discussed. Finally,
as an initial verification of the implementations, results from the low-order and CFD-based solutions are
compared for a rigid-airfoil rotor in forward flight.
Introduction
Reducing noise, vibration and power consumption
is becoming increasingly important to the modern
helicopter industry. This has inspired the development of many new technologies. Especially promising are those based on active blade control which
have a potential ability to adapt to flight conditions
as well as improve multiple characteristics through
the same system. Such concepts have traditionally achieved control without intentionally deforming the airfoil’s main supporting structure. Recently
Presented at the American Helicopter Society
65th Annual Forum, Grapevine, Texas, May 27-29,
c
2009. Copyright °2009
by the American Helicopter
Society, Inc. All rights reserved.
however, actuation systems have been proposed that
apply continuous shape changes through the use of
integrated active materials [1, 2]. These morphingtype systems may offer several benefits, including
improved aerodynamic efficiency as well as potential for good structural weight efficiency if the actuators themselves contribute to blade overall stiffness
and strength. Although some aspects of these systems have been previously studied, general design
requirements and effectiveness are still largely unknown.
In order to analyze morphing-type rotors in
depth, one must consider several phenomena normally assumed to be unimportant in rotor problems.
Structurally, shape changes can be introduced by a
combination of cross-section flexibility and actuator
forces. Bending along the chordline cannot be modeled in the beam formulations typically used for rotor analyses, while the coupled actuator/structural
dynamics may be difficult to analyze when the actuator is highly integrated. Aerodynamically, the possibility of general conformations adds a large degree
of complexity. Besides altering the behavior associated with linear phenomena, shape changes can also
signficantly change the effects of compressibility or
viscosity. To properly assess the impact on vibration and power, these higher-order effects must be
considered.
In the modern aeroelastic analysis of rotors, nondeforming cross sections are generally assumed in
the structural and aerodynamics theories, as well as
in their coupling. The structural theories are based
on classical beam degrees-of-freedom, and the aerodynamics use either a lifting line model (often augmented with semi-empirical models to account for
higher-order effects), or computational fluid dynamics. For the latter, the current state-of-the-art is
Reynolds-Averaged Navier Stokes (RANS). Examples of the successful coupling of structural dynamics analyses with RANS-CFD include CAMRADOVERFLOW [3], DYMORE-OVERFLOW [4],
RCAS-OVERFLOW [5], UMARC-TURNS [6], and
HOST-elsA [7]. Though significant progress has
been made in CFD-based aeroelastic simulation, the
ability to model deformable airfoils (with or without
actuation) while also having sufficient accuracy for
noise, vibration and power assessment has not been
developed to the authors’ knowledge.
This paper describes the development of a highfidelity simulation for the aeroelastic analysis of rotors with actuated, flexible airfoils. The simulation
consists of a RANS CFD analysis coupled threedimensionally with a structural analysis that captures camber deformations. The simulation is supported by a finite-state, semi-empirical aerodynamics analysis, which is used to assist initialization and
trim. The theories and coupling strategies of the
different formulations are presented. A preliminary
comparison of the aeroelastic response predictions
from the low-order and CFD-based analyses is also
shown for a rotor in forward flight.
Structural Dynamics Formulation
The computational structurual dynamics (CSD)
formulation used in the current study has been presented in Refs. 8 and 9. It follows the variationalasymptotic method for the analysis of composite beams [10]; that is, the equations of motion
for a slender anisotropic elastic three-dimensional
solid are approximated by the recursive solution
of a linear two-dimensional problem at each cross
section [9], and a one-dimensional geometricallynonlinear problem along the reference line [8].
This procedure allows the asymptotic approximation of the three-dimensional warping field in the
beam cross sections, which are used with the
one-dimensional beam solution to recover a threedimensional displacement field. The present implementation adds an arbitrary expansion of the displacement field through a set of functions approximating the sectional deformation field to capture
“non-classical” deformations, which are referred to
as finite-section modes.
The geometrically-nonlinear dynamic equations
of equilibrium along a reference line of a slender
solid were presented in Ref. 8. If all magnitudes are
expressed in their components in a reference frame
attached to the deformed reference line, they are
written as:
d
d
+ Ω̃B )PB = (
+ KB )(FB − f1 ) + f0
dt
dx
d
( + Ω̃B )HB + VB PB =
dt
d
+ KB )(MB − m1 ) + (ẽ1 + γ̃)FB + m0
(
dx
d
d
Qt =
(Qs1 − fs1 − (Qs0 − fs0 )
dt
dx
(
(1)
The first two sets of equations imply equilibrium
of forces and moments, where K is the current curvature of the reference line, the pairs (V, Ω) and
(P, H) are the local translational and rotational inertial velocities and momenta, respectively, and F
and M are the sectional forces and moments (similar
to traditional geometric-nonlinear blade modeling
in state-of-the-art rotorcraft aeromechanics codes).
In addition, f0 and m0 are the conventional (zeroorder) applied forces and moments, respectively, per
unit length on the beam, while f1 and m1 are the
first-order loads associated to the work needed to
deform the cross section. The last set of equations includes the equilibrium of the generalized
forces (Qs0 , Qs1 ) and momenta, Qt , corresponding to the finite-section modes, which are defined
by prescribed cross-sectional displacement shapes
ψq (x2 , x3 ) with amplitude q(x). These equations are
complemented by the cross-sectional constitutive relations, which, for a elastic solid with embedded ac-
tuation, are derived in Ref. 8 using the variationalasymptotic method. These equations take the final
form of

FB



MB
Qs0



Qs1

 PB
HB

Qt
 


γ





 


κ
−
= [S]



 
 q0 






q




 VB 
ΩB
= [M ]



q̇




F (a)
M (a)
(a)
Qs0
(a)
Qs1
h(ξ) =
Z





∞
X
hn Tn (ξ)
n=0
1
Ln = −b
Tn (ξ)∆P dξ
(3)
−1




(2)
where [S] and [M ] are the cross-sectional stiffness
and mass matrices, respectively. The superscript
(a) indicates actuation loads per unit length on the
structure, and γ and κ are the beam strain measures
associated to forces and moments, respectively. The
variational asymptotic method determines [S], [M ]
and actuation loads by discretizing the cross section
into finite elements, allowing an arbitrary geometry
and material distribution.
To develop a finite-element solution, the dynamics of the member are first described by a mixed
formulation in which the variation of the equations
of virtual work are taken with regard to displacement, force and momentum variables. These variables are used to define the structural state. Straindisplacement relations and velocity-displacement relations are satisfied simultaneously by using Lagrange multipliers. The resulting governing equations are first-order in time and space, which allows
the use of simple integration schemes.
The solution of these equations has been implemented in the computer code UM/NLABS (University of Michigan’s Nonlinear Active Beam Solver).
Low-Order Aerodynamics Formulation
Deforming Thin-Airfoil Theory
The low-order model uses the two-dimensional
finite-state formulation for flexible airfoils presented
in Ref. 11. It is based on a Glauert expansion of
the potential flow equations for a deformable airfoil
of infinitesimal thickness. The camber-wise airfoil
deformation is written as:
where h is the local displacement normal to the
chord, b is the semi-chord length and ξ is the nondimensional coordinate along the chordwise direction. Tn are the Chebyshev polynomials of the first
kind, which define the generalized camber-wise displacement amplitudes, hn , and its associated airloads, Ln integrated from the local airloads ∆P .
The final result (Ref. 11) for the airloads computed
in the normal and chord directions arising from the
potential flow assumptions is:
1
L̄
ρc`α
¨ + ν̄)
˙
= −b2 M (h̄
−bu0 C(h̄˙ + ν̄ − λ̄0 ) − u20 K h̄
−bG(u̇0 h̄ − u0 ν̄ − u0 λ̄0 )
1
D̄0
ρc`α
(4)
= −b(h̄˙ + ν̄ − λ̄0 )T S(h̄˙ + ν̄ − λ̄0 )
¨ + ν̄)
˙ T Gh̄
+b(h̄
−u0 (h̄˙ + ν̄ − λ̄0 )T (K − H)h̄
+(u̇0 h̄ − u0 ν̄ + u0 λ̄0 )T H h̄
L̄ =
ν̄ =
λ̄0 =
{L0 , L1 , Ln , ...}T
{ν0 , ν1 , 0, ...}T
{λ0 , 0, 0, ...}T
(5)
(6)
where L̄ is the vector containing the generalized airloads, and similar definitions are given for the generalized camber-wise displacements (h̄), rigid-section
normal velocity (ν̄), and section inflow distribution
(λ̄). The drag force is D0 , and M, S, G, K, and
H are constant coefficient matrices whose complete
definition is given in Ref. 11.
Dynamic Wake
The wake-induced velocity (λ0 in eqn. 6) is solved
using the dynamic inflow theory of Ref. 12. It assumes that the velocity normal to the rotor disk can
be expressed in terms of the radial and azimuthal
expansion functions,
w(s, ψ, t) =
∞
X
∞
X
φrj (s) ×
tion, the total airloads can be expressed as:
1
D0,tot
= D0 + ρbcd (u20 + ν̄ T S ν̄) 2 u0
L0,tot
L1,tot
= L0 − ρbcd (u20 + ν̄ T S ν̄) 2 S ν̄ + ρu0 Γ0
= L1 + +ρu0 Γ1
(10)
1
r=0 j=r+1,r+ψ,...
£ r
¤
αj (t)cos(rψ) + βjr (t)sin(rψ)
(7)
where s and t are the non-dimensionalized radius
and time, while ψ is the azimuth. The inflow states,
α and β, are coefficients of terms containing the
product of azimuthal harmonics and radial expansion functions φ, indexed by r and j, respectively.
The governing equations for the inflow states are
given by:
1 mc
{τ }
2 n
£ 1¤ r
1
M {β̇j } + [Ls ]−1 {βjr } = {τnms }
2
[M 1 ]{α̇jr } + [Lc ]−1 {αjr } =
(8)
The pressure coefficients, τ , are determined from
the circulatory lift, which is available from eqn. 4
by neglecting terms associated to accelerations in
the zeroth-order loading. The left hand side coefficient matrices, [M 1 ] and [Lc,s ] are determined by
the wake skew angle. Finally, with the solution of
the inflow states, the zeroth-order inflow (λ0 ) is calculated from:
µ ¶
∞
∞
X
X
rb
λ0 =
J0
φrj (s) ×
s
r=0 j=r+1,r+3,...
£ r
¤
αj (t)cos(rψ) + βjr (t)sin(rψ)
(9)
where J0 is the Bessel function of the first kind and
can be approximated by taking the first few terms
in its series.
Drag and Dynamic Stall
A potential benefit of camber actuation is the
ability to alter profile drag and stall characteristics,
which have implications in power and vibration. To
include these effects in the low-order model, the
potential-flow airload expressions are augmented
with a quasi-static profile drag term as well as a
dynamic-stall correction that is based on the ONERA model [13], taking the approach described in
Stumpf and Peters [14]. In the current implementa-
where cd is the profile drag coefficient, Γ0 and Γ1
are the dynamic stall states corresponding to the zeroth and first-order generalized loads, respectively.
The ONERA model assumes that the dynamic stall
states are governed by a second-order differential
equation, and requires static loading coefficients
near and beyond stall. These, along with cd , are determined using the two-dimensional boundary layer
analysis code XFOIL [15] (which is valid to justafter-stall), along with a simple, empirically-derived
approximation for deep-stall. The coefficients are
obtained under varying Reynolds number, angle-ofattack and camber deformation. A detailed account
of the method used for determining the coefficients
is available in Thepvongs et. al [16].
Coupling with Finite-State Aerodynamics
The finite-state aerodynamics formulation uses
Chebyshev polynomials to form a basis for the camber deformations and associated airloads, while the
choice of basis functions for the finite-section modes
are arbitrary. The motion and force variables of the
aerodynamics formulation are related to those of the
structural formulation by a simple linear expression,
as derived in Thepvongs et. al [16]. This straightforward connection between the aerodynamic and
structural states allows the same space and time integration methods to be used for both formulations
as well as a simultaneous solution.
The governing structural dynamics equations
(eqn. 1), aerodynamic load expressions (eqns. 4, 5),
dynamic stall equations and wake equations (eqn. 8)
together define the time-domain problem. An explicit method is used with iterative refinement to
achieve the desired convergence. Consider the following form of the structural dynamics equations,
obtained by applying spatial discretization (finiteelements) to eqn. 1. The set of differential-algebraic
equations, along with the appropriate set of boundary conditions can be written as:
A(Xp )Ẋp + S(Xp , X̂p )
BC(X̂p )
= LF S (Xp , Ẋp , X̂p ,
α |p , β |p , Γ̄p )
= 0
(11)
where A is the inertia matrix operator, S is the
structural matrix operator, written as functions of
the structural state vector, X , its time derivative,
Ẋ, and boundary values, X̂. These, along with the
aerodynamic states, are all obtained at the current
time (time is indexed by p).
The load matrix operator LF S is the contribution of the aerodynamic loading which is a function of the structural, inflow, and dynamic stall
states. A simple three-point backwards Euler timeintegration scheme is used in accordance with the
first-order form of the structural and potential flow
¨ term of eqn. 4 is obtained
governing equations (the h̄
by first derivatives of the velocity, which is a structural state variable due to the mixed formulation).
A four-point scheme is used to integrate the secondorder dynamic stall equations. The time-integration
formulas are:
˙ = 3()p − 4()p−1 + 1()p−2
()
p
2∆t
2()
−
5()
+
4()
p
p−1
p−2 − 1()p−3
¨ =
()
p
(∆t)2
τ θ̈ + θ̇
=
Jkl
≈
J −1 {a(G − g) − bġ − dg̈}
Z
1 T ∂gk
dt
T 0 ∂θ1
where θ = [ θ0 θ1c θ1s ]T is the control settings,
and G and g are the target and current values of the
variables representing equilibrium (i.e., the thrust,
pitch and roll moments), respectively. For the test
cases run in this study, coefficients b, d, and τ are
set to zero, and a = 1/T , where T is the period.
The ”trimmability matrix” J can be approximated
by numerically-computed Jacobian, determined by
stepping the controls and examining the response
at an instant one revolution later. It should be
noted that the trim is solved externally to the aerodynamic and structural equations, using the time
integration scheme of eqn. 12. Since the body loads
are computed from structural variables, namely the
root reaction loads, the trim is independent of the
aerodynamics. Thus, the same trim algorithm and
implementation is used for the low-order and CFDbased analyses.
High-Fidelity
Formulation
(12)
The nonlinear system is solved, along with the
boundary conditions of eqn. 11 by Newton-Raphson
iterations, with the Jacobians of the left-hand side
operators of eqn. 11 available in closed form.
Trim
The enforcement of vehicle equilibrium adds more
variables and constraints to the aeroelastic problem.
The present work assumes a wind-tunnel setup,
where the variables are taken to be the collective,
sine and cosine components of the cyclic pitch, and
equilibrium is represented by specifying values for
the time-averaged thrust, pitch and roll moments.
An auto-pilot method has been described by Peters
et. al [17], which makes incremental changes to the
control settings at every timestep. The control settings are governed by:
(13)
Aerodynamics
Unstructured or mixed-element CFD methods
have advantages over structured-based CFD methods that can be exploited for rotorcraft analysis. In
particular, rapid grid generation for complex configurations is usually cited as a major reduction in
the preprocessing stage. Additional advantages exist because unstructured methods can model rotors
with fewer overset grids, resulting in less computational expenditure for grid motion and potentially a
reduction in numerical errors via fringe and orphan
computations. The most physically correct analysis requires that all surfaces be accurately modeled via the time-accurate Navier-Stokes equations,
requiring that two frames of motion be modeled
during a single simulation. While several options
exist, one of the most applied approaches utilizes
a combination of overset grids to generate smaller
grids around each rotor blade, which then rotate
through a background grid that may include a fuselage, ground planes, or other important configurations. This overset approach has been used to correctly capture the general experimental trends on
several configurations by Hariharan [18], Potsdam et
al [3] and Duque et. al [19], O’Brien and Smith [20]
and Abras and Smith [21].
Because of the potential of the unstructured
methodologies for advanced rotor configurations, as
well as their ability to rapidly model rotor vehicle
components, an unstructured methodology has been
chosen to provide the aerodynamic simulation for
this effort. The unstructured methodology selected
for this effort is the FUN3D code developed at the
NASA Langley Research Center [22–24]. FUN3D
can resolve either the compressible or incompressible RANS equations on unstructured tetrahedral or
mixed element meshes. The incompressible RANS
equations are simulated via Chorin’s artificial compressibility method [25]. A first-order backward Euler scheme with local time stepping has been applied
to steady-state applications, while a second-order
backward differentiation formula (BDF) has been
utilized for time-accurate simulations. A pointimplicit relaxation scheme resolves the resulting linearized system of equations. The RANS equations
are resolved via non-overlapping control volumes
surrounding each cell vertex or node where the flow
variables are stored. Inviscid fluxes on the cell faces
are solved via Roe’s scheme [26] that splits the flux
differences. The viscous fluxes are computed with
a finite volume formulation to obtain an equivalent
central-difference approximation.
The FUN3D methodology was originally developed for fixed-wing applications by NASA researchers. Georgia Tech researchers have extended
it for rotary-wing applications of interest, as described by O’Brien and Smith [20] and Abras and
Smith [21]. O’Brien [27] has shown that FUN3D’s
steady incompressible formulation is not only robust
for low-speed flight regimes, but both compressible
and incompressible results comparable to structured
methods are obtained when the grids are locally
comparable on the surface and boundary layers. In
addition, O’Brien developed an overset formulation
of FUN3D to handle mixed inertial-moving frames
necessary to model rotating wings. This formulation
makes use of the DiRTLib [28] and SUGGAR [29]
libraries to compute the cell connectivity and provide the hole cutting necessary with moving frames.
Abras [30] has extended the FUN3D code further
to provide full rotor articulation, as well as loosecoupling with a computational structural dynamics
code, assuming that the rotor is modeled as a structural beam. These extensions form the basis for the
three-dimensional aeroelastic rotating wing simulations, extended further to include camber deformations.
Modifications for Aeroelastic Coupling
While many of the components necessary to perform rotating, overset computations were extended
by prior Georgia Tech researchers, there remained
a number of modifications that were necessary to
accomplish full surface deflections within a tightlycoupled framework. FUN3D is capable of handling
multiple grids to define the rotor blade and, as a
massively parallel simulation methodology, will decompose each grid further as it parses the computational load among a number of processors. The current coupling scheme uses subdivisions of the blade
surface nodes, dividing them into smaller groups (as
later described). In order to couple FUN3D, the
nodes of each of the blade surfaces in FUN3D must
be identified and sorted into the groups. The blade
local coordinates of the surface nodes in FUN3D are
collected from each processor. The CFD nodes that
define the edges of the groups may belong to more
than one group, so the nodes are flagged to ensure
that the same surface node is not passed to included
in the coupling more than once. Another array containing the coordinates of the surface face centers is
created and sorted in the same way.
As is the case with most CFD solvers, FUN3D
resolves the primitive variables of the flowfield using a nondimensional system. NLABS, as is typical
for CSD solvers, uses dimensional quantities. Thus,
the pressure coefficient on each cell face that lies on
the blade surface is computed within the CFD code,
and are then dimensionalized using the freestream
pressure of the simulation. Similarly, spatial coordinates of each cell are dimensionalized and the area
of the CFD cell computed, so that a dimensional
force is computed from the cell pressure and area.
This force array is passed into NLABS, which finds
the structural deflections at each node location and
passes the new coordinates back to FUN3D, along
with the orientation of the blade.
For the coupling, it is necessary to deform not
only the CFD surface, but also the CFD volume
grid. As illustrated in Refs. 21 and 30, large surface deflections from not only the elasticity of the
blade but new trim or articulation commands can
cause the blade near-body grid to encounter poor
cell resolution. In order the minimize the volume
grid deformation, the elastic deflection of the blade
is included in the Euler angle rotations. For example, in this application, the pitch angle is defined
as the control pitch angle plus 0.5 times the elastic
twist of the blade at the tip. Flap and lead/lag angles are defined using a straight line extending from
the elastic axis at the root to the location of the
elastic axis of the tip. Using the deformations and
any control changes from NLABS, the pitch, flap,
and lead/lag angles are extracted and updated in
the six degree-of -freedom motion equation. The remaining elastic deformations are then applied to the
rotor surface. The near-field volume grid based on
the new surface node coordinates is then generated,
and the SUGGAR library [29] is then accessed to
calculate new connectivity data. At this point the
flow solver is allowed to advance to a new time step
at which point the process is repeated.
Figure 1: Simulation flow.
Time Coupling and Initialization
loads are set to gradually transition from those of
the low-order model to those of the CFD in order
to prevent disturbances that may be slow to decay.
Due to its computational cost, the CFD-based
analysis uses a weaker coupling of the fluid and
structural solutions compared to that of the loworder model. Any coupling scheme that needs repeated solutions of the CFD or modification of the
CFD boundary conditions during a timestep would
not likely be affordable. Instead, the current approach obtains separate solutions for the CFD and
CSD analyses, where, within a given timestep, the
computed state of one is held fixed during the solution of the other, using the most-recently available information and no subiterations. It has been
demonstrated by prior work [31, 32] that for hover
and forward flight that a loose-coupling between the
CFD and CSD codes is sufficient. Further, fixedwing researchers, (e.g. Refs. 33, 34), have shown
that for the small time step size needed for a stable
(uncoupled) CFD solution, that the resulting errors
are negligible.
To further reduce computational time, the loworder aerodynamics model supports the initialization and trimming of the CFD-based aeroelastic
analysis. Long-time simulations are required for
computation of the trim Jacobian and convergence
to a sufficiently periodic and trimmed response. The
low-order model can be used to compute the Jacobian and provide a starting condition of the CFDbased analysis. The starting condition is obtained
by allowing the low-order model to run to convergence, then extracting the structural state and
control settings. These are used to initialize the
CFD-based analysis, which is run with the low-order
model kept activated. The applied aerodynamic
The time-coupling scheme described above is formalized by augmenting eqn. 11 so that the scheme
becomes:
A(Xp )Ẋp + S(Xp , X̂p ) =
ζLCF D (Xp−1 , Ẋp−1 , X̂p−1 )
+(1 − ζ)LF S
(14)
where LCF D represents the numerically-computed
(non-linear) CFD loads (transformed and integrated
to structural forcing variables, as described in the
following section) and ζ is the weighting factor that
controls the transition between the CFD and loworder loads (0 ≤ ζ ≤ 1).
A diagram showing the general operation of the
combined aeroelastic simulation is shown in Fig. 1.
Transformation of
Structure Variables
Fluid
–
The fluid and structure wetted surfaces are discretized separately according to the resolution requirements of the individual analyses. Therefore
a transformation is needed to properly connect
the motion and force variables between the two.
This transformation has to consider the presence
of camber-wise flexibility in addition to the usual
beam degrees-of-freedom. Although the finite section modes are treated as one-dimensional beam
variables, they are defined in a discrete way within
the cross-section analysis. This is accomplished by
specifying the modes’ contribution to the displacement at a selected number of grid points within
the cross section (which are a subset of the total grid points used to discretize the cross-section
problem). Additionally, the motion of these crosssection grid points is determined partly by warping
arising from sources other than the finite-section
modes. For these reasons, quantities at the CFD
grid cannot be directly expressed by (or integrated
to) one-dimensional beam variables, as is often done
when coupling CFD to conventional beam analyses [31, 32].
An intermediate step involving the threedimensional structural grid is required. From this
three-dimensional structural grid, the displacements
are interpolated to the CFD grid finite-element
shape functions, as originally described by Pidaparti [35] and extended by [37]. A triangulation
of the structural grid points forms a new set of elements which are used solely for the interface. Next,
for each structural interface element, all enclosed
CFD nodes are identified. The criterion for enclosure is that there exists a vector normal to the
element that intersects both the element and the
grid point. Among the structural interface elements
meeting this criterion, the one having the shortest
distance to the CFD node is used for interpolation.
D
The displacement of this CFD node, uCF
, is dej
fined by:
D
uCF
= Nj uCSD
+ Rdj
j
i
(15)
uCSD
i
is the column matrix of the displacewhere
ments at the nodes of the structural interface element i, and Nj is the row matrix containing the element shape functions evaluated at the CFD node.
The shape functions are linear for the 3-node element. The last term in the above equation accounts for separation between the CFD node and
the element plane. It preserves the separation distance and CFD node’s local coordinate within the
element throughout the deformation. The vector
from the element plane to the CFD node is dj , and
the matrix rotating the element normal from its undeformed to deformed orientation is R.
The transformation of the forces from the CFD
cell centers to the structural interface mesh uses the
same finite-element shape functions. The force at a
CFD cell, FkCF D , is distributed to the nodes of the
enclosing structural interface element by the relation:
FiCSD = NkT FkCF D
(16)
where Nk represents shape functions evalutated
at the cell center position k. Some CFD nodes may
not be enclosed by a structural interface element
due to the boundaries of the CFD and CSD grids
being slightly offset. The present study attempts
to model the same geometry (the wetted surface) in
each analysis, therefore the offset is caused only by
the different discretizations in each grid or numerical precision errors. For the displacement of these
points, extrapolation is performed using a function
fitted to the structural interface grid:
D
=
uCF
j
P
X
£
¤1
D 2
− xCF
) + r2 2
γji (xCSD
i
j
(17)
i=1
D
are the position of the strucwhere xCSD
and xCF
i
j
tural and fluid nodes, respectively. P is the total
number of structural nodes over which the extrapolation function is formed, r is a shape parameter
(Ref. 36) and γji are the computed weighting coefficients. Following the extrapolation, the displacements of these points are averaged with its neighbors to improve smoothness between interpolated
and extrapolated regions. Forces originating from
CFD cell centers not enclosed by any element are
simply lumped along with its moment contribution
to the nearest structural node.
In the current study, the CSD and CFD meshes
are subdivided into four domains, corresponding to
the tip, root, top and bottom surfaces of the blade.
The interpolations (specifically, the searching) and
extrapolations are done separately for each domain
to reduce overall computational costs. The interpolation and extrapolation algorithms described above
are implemented in modified versions of the work by
authors of Refs. 36 and 37.
It should be mentioned that the fluid and structural problems should ideally form a closed system,
as discussed by Smith [38]. The right-most term of
eqn. 15, which has been added as part of the current
work, removes the conservation of work across the
interface that normally occurs with finite-elementbased interpolation. This term, however, improved
accuracy of the interpolations when the separation
distance between the CFD nodes and its enclosing
element became large, as discussed in the next section. Similarly, the method of extrapolation and
integrating forces for unenclosed CFD nodes does
not enforce conservation of work. Errors resulting
from the addition/loss of energy through the transformations are presumed to be small. This should
be further investigated, however, in future studies.
Numerical Investigation: Rotor
Model and Discretization
Results presented in this study are based on a
scaled BO105 rotor model. The rotor is hingeless,
4-bladed, and has a 2-m tip radius. From 0.44R
to the tip, the blade has a 0.121-m chord and an
NACA 23012 airfoil, modified with a trailing-edge
tab. The rotation speed is 109 rad/s.
The CFD blade surface grid (Fig. 2) was constructed from the detailed geometry provided in
Ref. 39. The moving-volume grid forms a rectangular prism that encloses the region between 1-chord
length inboard of the root-most blade edge to 2chord lengths beyond the tip, 1-chord length ahead
of the leading edge to 1-chord length behind the
trailing edge, and 2-chord lengths below the blade
to 2-chord lenghts above it. A total of 896,241 nodes
are contained in the moving-volume grid of each
blade. The background grid is a cylinder extending
40 m above and below the rotor, with a diameter of
80 m, and is comprised of 1,046,807 nodes.
The CSD discretization is defined by the distribution of finite-element nodes along the beam reference line, along with grid points within the crosssections at those nodes. For the reference line, there
are 43 elements, concentrated near the transition region. Each cross section contains 80 grid points, distributed along the wetted surface and concentrated
near the leading and trailing edges. These grid
points and their triangulation for the fluid-structure
interface is shown in Fig. 3. A summary of the fluid
and structural discretizations is given in Table 1.
For the cross-section properties, integrated mass
and stiffness data for the conventional beam
degrees-of-freedom were available, however, detailed
layup information could not be obtained. The finitesection mass and stiffness were set to values [16]
that prevented significant camber deformation due
to aerodynamic loads or dynamic response. This
created a realistic condition for the initial validation of the simulations.
(a) CFD surface grid
(b) Near-field blade grid
(c) Full CFD volume grid
Figure 2: CFD grids for the BO105 rotor.
Figure 3: CSD mesh for the BO105 rotor.
Table 1: Blade discretization summary (per blade).
CFD surface nodes
CFD surface cells
Structural interface nodes
Structural interface cells
Structural 1-D elements
Figure 4: Test load magnitude (N ) at discrete
points on CFD grid.
20398
40972
3680
6864
43
Numerical Investigation:
sults
Re-
Verification of the Fluid – Structure
Transformation
The basic methodology of the interface algorithms
has already been validated for a set of canonical
cases, as presented by Lee [37]. Several aspects of
the interface are re-examined, using the grids for
the BO105 model currently being studied. To verify
the transformation of forces, a load was prescribed
at the CFD cell centers that varied substantially in
amplitude and orientation. Fig. 4 shows the distribution of the amplitude along the top and root
surfaces. These forces were then transferred to the
structural nodes through the interface, and integrated to find the total forces and moments acting
on the blade (the latter about the hub center). Under this loading, the error in the global forces along
the three axes was found to be less than 1 × 10−12 %
and for the moments, less than 0.01%.
A visual approach was employed to verify the
displacement transformations. A sample displacement field was set up by applying transverse and
finite-section (camber-wise bending) forces along
the structure, producing large displacements. Fig. 5
shows the co-movement of the fluid and structural
wetted-surface nodes.
The necessity of the rotational term in the in-
Figure 5: Sample undeformed and deformed grids
including camber-wise bending response – small
points: CFD; large points: structural interface.
terpolation scheme of Eqn. 15 is demonstrated in
Fig. 6. Because the CSD grid is substantially coarser
than that of the CFD, the CFD-node-to-interfaceelement distance can become large when there is
discontinuous geometry in the CFD grid. An example is the transition region of the blade, near
the trailing-edge tab. Fig. 6 shows this region,
where the view is from the root towards the tip,
as indicated by the arrow in Fig. 2(a). A 20◦
rigid-body pitch rotation is applied to the structure, and the interface transforms the CFD mesh
shown in Fig. 6(a). The rotational term in the interpolation equations preserves the shape of the mesh
(Fig. 6(b)), while neglecting the term causes the surface to become wrinkled (Fig. 6(c)).
Forward Flight
With verification of the fluid-structural interface,
several characteristics of the aeroelastic response are
examined for a forward-flight condition. The advance ratio considered is 0.25, and the shaft angle
is −5◦ . Following the wind-tunnel experiments of
Ref. 39, the trim condition is 3100N thrust (Fz ),
zero pitching moment (My ) and zero rolling moment
(Mx ), evaluated as time-averaged quantities. In the
first set of results, the effect of the transition length
between the low-order and CFD loads is examined.
Three time functions are setup for the variation of
(ζ) in Eq. 14. The first is a step function, while
the second two increase linearly from 0 to 1, over a
half and full revolution. In the following figures, the
first revolution contains the response computed by
the low-order analysis after its convergence to periodicity and trim. After this revolution, the CFD
solution begins along with the transition from the
low-order to CFD loads, using one of the three transition methods.
(a) Original
(b) With rotational term
(c) Without rotational term
Figure 6: CFD grid near trailing edge of transition
region; a) original; 20 deg. pitch applied through
fluid-structure interface with (b) and without (c)
rotational term in Eqn. 15.
Figure 7 shows the time-history of the instantaneous, fixed-frame hub loads, while Fig. 8 shows the
control settings. Comparing the converged control
settings of the low-order and CFD-based analyses
(Table 2), there is a 0.6◦ difference in the collective,
while the cosine and sine components differ by 0.2◦
and 0.3◦ , respectively. The step transition from loworder to CFD loads produces the largest deviation
from the previously-established equilibrium. This
deviation is reduced by the half-revolution ramp,
then further by the full-revolution ramp. Despite
this perturbation, the convergence of both the control settings and hub loads appear to take less time
overall using the step transition. In examining the
control settings, the collective changes at a faster
rate with the step transition while also appearing
to have a head start compared to the other transition methods. The sine and cosine components
of the pitch control also show earlier arrival at the
converged value, although the rate appears to be
unaffected by the transition method. These results suggests that allowing the trim algorithm to
act earlier on the CFD-based aeroelastic response is
more important than avoiding disturbances caused
by sudden transitioning. In other words, the damping (which is dominated by aerodynamic loading) is
sufficient to decay the disturbances such that they
are insignificant well before trim is reached.
Table 2: Control settings in degrees at convergence
of trim and aeroelastic response.
θ0
θ1c
θ1s
The
second
CFD
10.8
0.7
-2.5
set
of
Low-order
10.2
0.857
-2.82
results
compare
the
CFD
Low−order
x
F (N)
100
θ1s (deg.)
0.5
1.0 revs.
F (N)
y
10.5
10
1
Fz (N)
0
−2.5
−3
0
1
2
0.0
3
4
Revolutions
5
6
7
0.5
1.0 revs.
Fz (N)
3500
Mx (Nm)
3100
3000
Figure 7: Pitch control history, varying length of
transition between low-order and CFD loads.
3000
2500
2000
100
0
−100
−200
My (Nm)
0
−200
3200
0.5
−3.5
0
−100
200
11
θ1c (deg.)
θ0 (deg.)
0.0
0
−100
−200
0
1
2
3
4
Revolutions
5
6
7
Figure 8: Hub loads history, varying length of transition between low-order and CFD loads.
0
0.25
0.5
Revolutions
0.75
1
Figure 9: Hub forces comparison of converged loworder and CFD predictions.
periodically-converged, trimmed responses of
the low-order and CFD analyses as an attempt to
co-validate them. The fixed-frame hub forces are
shown in Fig. 9, while the moments are shown in
Fig. 10. The rearward and starboard-side loads,
Fx and Fy , show good agreement in frequency and
phase of the response, though significant differences
are seen in the amplitudes, as well as the mean
value of Fy relative to its amplitude. As expected,
the mean values of Fz , Mx , and My approach the
values enforced by the trim algorithm, however,
large differences can be seen in most other aspects
of the time response. Also, in the torque (Mz ), the
low-order prediction of the magnitude of the mean
value is 13% smaller than that of the CFD which
if uncoupled from the other errors, may indicate
that improvements in the drag model are needed.
Lastly, it is important to note that for all of the
fixed-frame loads, the CFD generally predicts a
larger vibratory amplitude.
The tip deflections are presented in Fig. 11. The
chordwise (positive in the leading direction), flapwise (positive up) and pitchwise (positive nose up)
responses are shown. Several similarities can be
seen in the predictions from the two analyses. In
the chordwise response, both results show a dominant 1/rev component and a similar phase. The
CFD loads produce a larger mean deflection in the
lag direction, which agrees with the larger torque
magnitude found in the hub loads. The flapwise response also shows agreement in several aspects. The
positive peak occurs near 270◦ azimuth (0.75 of the
Low−order
CFD
flapwise (m) chordwise (m)
CFD
Mx (Nm)
100
0
0
−50
−300
−350
−400
−0.015
−0.02
pitch (deg.)
Mz (Nm)
My (Nm)
−100
50
0
0.25
0.5
Revolutions
0.75
1
Low−order
0.08
0.06
0.04
6
4
2
0
−2
0
0.25
0.5
Revolutions
0.75
1
Figure 10: Hub moments comparison of low-order
(converged) and CFD (latest rotation available) predictions.
Figure 11: Tip deflections comparison of low-order
(converged) and CFD (latest rotation available) predictions.
revolution), while a flat response is seen near 0◦ azimuth. The CFD loads, however, appear to induce
a more significant 3/rev component than those of
the low-order model. The mean values from both
predictions are similar. In the pitchwise response,
the absolute rotation is shown (includes both elastic
and rigid-body contributions). The CFD predicts a
more negative overall rotation at the tip, despite
the increased collective pitch. This suggests that
a larger nose-down aerodynamic pitching moment
is computed by the CFD analysis, and this may be
the source of the requirement for increased collective
pitch.
which has been parallelized, is pending and should
improve the execution speed. The combined loworder aerodynamics, structures and fluid-structure
transformations take several seconds of serial computations for each time step.
Despite the differences in the aeroelastic response
predicted by the two aerodynamics models, correlation in several aspects including control settings,
tip flapwise response and some hub loads, appear
to verify a correct implementation, especially given
the vastly different approaches used in coupling the
analyses.
Computational Requirements
The coupled code is executed on a cluster of IBM
Power5+ 1.9 GHz processors. For the high-fidelity
analysis, a total of 64 processors have been used:
63 for the parallel FUN3D computations and one
for the serial computations which includes NLABS
and SUGGAR. Execution takes about 4.5 minutes
per time step (about 27 hours per revolution). It
should be noted that a new version of SUGGAR,
Concluding Remarks
A simulation for high-fidelity analysis of rotors
with actuated, flexible airfoils has been presented.
It includes a quasi-three-dimensional CSD analysis,
RANS CFD (FUN3D) analysis, as well as coupling,
trim and initialization algorithms. Results were presented for a scaled BO105 rotor that explored several aspects of the coupling in addition to the general time response. When using a low-order aerodynamics model for initializing the structural state
and control settings, a sudden transition between
the low-order and CFD-based loads was found to
produce faster convergence to a periodic, trimmed
solution compared to that produced by a gradual
transition. A comparison of the aeroelastic response
predicted by the low-order aerodynamics model and
CFD showed agreement in trimmed control settings,
as well as aspects of the tip deflections and fixedframe hub-loads. Although further validations are
needed, these results serve as a basic verification
of the methodology for coupling FUN3D with the
three-dimensional CSD analysis.
Acknowledgments
This work is sponsored by the National
Rotorcraft Technology Center (NTRC) Vertical
Lift/Rotorcraft Center of Excellence (VLRCOE) at
the Georgia Institute of Technology and University
of Michigan. Dr. Mike Rutkowski is the technical monitor of this center. Opinions, interpretations, conclusions, and recommendations are those
of the authors and are not necessarily endorsed by
the United States Government. Computational support for the NTRC was provided through the DoD
High Performance Computing Centers at ERDC
and NAVO through HPC grants from the US Army
(S/AAA Dr. Roger Strawn).
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