NASA Technical Memorandum 102733

AEROELASTIC ANALYSIS OF WINGS USING THE EULER

EQUATIONS WITH A DEFORMING MESH

BRIAN A. ROBINSON
JOHN T. BATINA
HENRY T. Y. YANG

(NASA-T"-lO2733) AEKnEI_AS]rlC A,ALYSI _ CF t_91-1007,_

t,/I/'qG_/ USING THE EUL_R =OUATTL_JS _[TH A

i2FF_MING M_SH (NASA) iI p CSCL OIA

Unc 1 os
r,3/o2 0310_30

NOVEMBER 1990

N/ A
Nalional Aeronautics and
Space Administralion

Langley Reseerch Cenlm'

Hamplon. Virginia 23665
J, o
 AEROELASTIC ANALYSIS OF WINGS USING THE EULER EQUATIONS
WiTH A DEFORMING MESH

Brian A. Roblnaon"
McDonnell Aircraft Company
St. Louis, Misaourl 63166

John T. Batlna*"
NASA Langley Research Center
Hampton, Virginia 23665-5225

Henry T. Y. Yangt
Purdue University
West Lafayette, Indiana 47907

Abstract ki generalized stiffness of mode i

Modifications to the CFL3D three-dimensional unsteady
Euler/Navier-Stokes code for the aeroelastJc analysis of m airfoil mass per unit span
wings are described. The modifications involve including a
deforming mesh capability which can move the mesh to mi generalized mass of mode i
continuously conform to the instantaneous shape of the
aeroelastically deforming wing, and including the struclural M. freestream Mach number
equations of motion for their simultaneous time-integration
with the governing flow equations. Calculations were
qi generalized displacement of mode i
performed using the Euler equations to verify the
modifications to the code and as a first-step toward
Qi generalized force in mode i
aeroelastic analysis using the Navier-Slokes equalions.
Results are presented for the NACA 0012 airfoil and a 45 °
sweptback wing to demonstrate applications of CFL3D for
generalized force computations and aeroelastic analysis. Q dynamic pressure, _--P.
Comparisons are made with published Euler results for Ihe
NACA 0012 airfoil and with experimental flutter data for the
nondimensional dynamic pressure, (U./(bo_.V'_p))2
45 ° sweptback wing to assess the accuracy of the present
capability. These comparisons show good agreement and,
thus, the CFL3D code may be used with confidence for ra airfoil radius of gyration about elastic axis
aeroelastic analysis of wings. The paper describes the
modifications that were made to the code and presents results t time, nondimensionalized by lreestream speed ol
and comparisons which assess the capability. sound and airfoil chord, Ta.Jc

T time, seconds
Nomenclature
ui load vector
nondimensional distance from midchord to elastic
axis U. sireamwise freestream speed

a= freeslream speed of sound

xi stale veclor

Aij generalized aerodynamic force resulting from x= nondimensional distance from elastic axis
pressure induced by mode J acting through mode i to mass center

b semichord, c/2
c=
o mean angle of a.ack
c reference chord
qB integral of state-transition matrix

ci generalized damping of mode i

ix mass ralio

cp pressure coefficient
p. froestream density

k reduced frequency, _c/2U.

a real part of Laplace transform variable
*Engineer-Technology, Aerodynamics, CFD Project;
formerly, Graduate Research Assistant, School of slate-transition malrix
Aeronaulics and Astronaulics, Purdue University;
Member AIAA. angular frequency
"'Research Scientist, Unsteady Aerodynamics Branch,
Structural Dynamics Division, Senior Member AIAA. mh uncoupled natural frequency of bending mode
1" Professor and Dean, Schools of Engineering, Fellow
AIAA. _.. uncoupled natural frequency of torsion mode
 Introduction allow simultaneous time-integration with the governing flow
In recent years, there has been increased interest in the equations for aeroelastic analysis. Calculations were
development of aeroelastic analysis methods involving performed using the Euler equations In verify the
computational fluid dynamics techniques. 1 This research has modifications to the code and as a first step toward
been highly focused on developing finits-diffarence codes for aeroelastic analysis using the Navler-Slokes equations.
the solution of the transonic small-disturbance2, 3 and full Results are presented for the NACA 0012 airfoil and a 45"
sweptbeck wing to demonstrate applications of CFL3D for
potential4, 5 equations, although efforts are currently
generalized force computation and aeroelastic analysis.
underway at the higher equation levels as well. 6"11 For Comparisons are made with the Euler results of Ref. 11 for
example, Bendtksen and Kousen 6 presented transonic flutter the NACA 0012 airfoil and with the experimental flutter data
results for two-degree-of-freedom (plunging and pitching) of Ref. 15 for the 45" sweptback wing to assess the accuracy
airfoils by simultaneously integrating the structural of the present procedures. The paper describes the
equations of motion with the two-dimensional unsteady Euisr modifications that were made In the CFL3D code and presents
equations. The Euler equations were Inlegrated using a results and comparisons which assess the capability.
Runge-Kutta time-stepping scheme involving a finite-
volume spatial discretization and a moving mesh. The
instantaneous mesh was taken to be a linear combination of Euler Solution Alaorithm
meshes corresponding to rigid plunging and pitching of the In the present study the flow was assumed to be governed
airfoil. In a following study, Kousen and Bendiksen 7 applied by the time-dependent Euler equations which may be written
their method of Ref. 6 to investigate the nonlinear in conservation form as
aeroelaslic behavior of two-degree-of-freedom airfoils at
transonic speeds. In Ref. 7, transonic flutter instabilities
were shown to lead to stable limit-cycle oscillations. Wu,
o
(1)
Kaza, and Sankar 8 have time-integrated the unsteady
compressible Navier-Stokes equations for airfoils where the vector _ represents the conserved variables
undergoing one- and two-degree-of-freedom eeroelastic
motions. In Ref. 8, flutter characteristics of airfoils at high divided by the Jacoblen and I_, _, A are the inviscid fluxes
angles of attack were investigated including cases with stall which have been transformed from the cartesian (x, y, z)
flutter. The method of Ref. 8 has also been recently applied coordinate system to generalized (_,,'q,_)coordinates.
by Reddy, Srivastava, and Kaza 9 to study the effects of Equations (1) are solved within the CFL3D code by a three-
rotational flow, viscosity, thickness, and shape on the factor, implicit, finite-volume algorithm based on upwind-
transonic flutter dip phenomena. The study concluded that biased spatial differencing. The upwind-biased differencing
the influence of these effects on flutter, for the cases involves either flux-vector splitting or flux-difference
cons_oereo, was small near the minimum of the flutter dip, splitting implemented as a cell-centered disoretization.
but may be large away from the dip. Guruswamy 10 has Flux-limiting may also be used in the spatial differencing to
demonstraled the simultaneous time integration of the three- determine values of the flow variables on the cell faces. For
dimensional Euler equations along with the structural unsteady applications, the algorithm includes the grid speed
equations of motion. The capability was demonstrated in a metric terms that are necessary for time-accuracy with
time-marching flutter analysis performed for a rectangular moving meshes, although the original scheme was limited to
wing with a parabolic-arc airfoil section. Finally, Rausch, cases involving rigid-body plunge or pitch where the mesh
Batina, and Yang 11 have presented Euler aeroelastic results moves without deformation. Modifications In the algorithm
for two degree-of-freedom airfoils using a flow solver based to include the lerms adsing from a deforming mesh, which
are required for aeroelastic analysis, are described in the
on unstructured gdds. A novel aspect of the capability of
following section.
Ref. 11 is the dynamic mesh algorithm which is used to move
the mesh so it continuously conforms to the instantaneous
position of the airfoil. The algorithm is completely general Deformina Mesh Alnorilhm
in that it can treat realistic motions and deformations of
multiple two-dimensional bodies. A deforming mesh algorithm was developed and
implemented to move the mesh so that it continuously
Many of the methods that are currently being developed conforms to the instantaneous shape of the aeroelastically
for aeroelastic analysis assume that the mesh moves rigidly deforming wing. The method, based on thal of Ref. 14,
or that the mesh shears as the body deforms. These models the mesh as a spring network where each edge of each
assumptions consequently limit the applicability of the hexahedral cell represents a linear spdng. The stiffness of
procedures to rigid-body or small-amplitude molions. each spring is inversely I:X'oportional to a specified power of
These problems, for example, are easily demonstrated by the length of the edge. For example, along an edge
considering an airfoil section of a wing whose aeroelaslic
deformation involves significant chordwtse bending. As the (i)- (i +1), the stiffness k is
airfoil section bends, grid lines that eminate from the
concave surface of the section may collapse onto the
neighboring grid lines. Similar difficulties can occur if the
wing has significant spanwise bending. Also, for methods k =1.01 [ (x , _ x)=. (YI+,-Yl) =+(Zl.,-Z) =lp'=
J
where the mesh is constrained such thai airfoil sections of 0+_1 +
2
the wing can only pitch and plunge, chordwise deformation (2)
cannot be modeled and wing tip deflections must remain
small. Therefore the purpose of the paper is to describe the In addition to the cell edges, springs are also placed along the
implementation of a deforming mesh capability which diagonals of each cell face to control cell shearing, with
effectively removes the rigid-body and small-amplitude spring stiffness defined similar to Eq. (2). The power p
limitations of previous methods. This capability was which appears in Eq. (2) is used to control the stiffness of
developed within the CFL3D unsteady EulerlNavler-Stokes the cells near the wing. These cells are typically very small
code. 12.13 The deforming mesh capability is a general in comparison with cells in the far field, and as such it has
procedure, based on the dynamic mesh algorithm of Refs. 11 been found advantageous to increase the stiffness of the near
and 14, which can move the mesh for realistic motions and field cells to avoid excessive mesh distortion in this region.
structural deformations of wings. In addition, the structural The stiffness is increased by increasing the value of p. In
equations of motion have been implemented within CFL.3D to this study, p was typically set equal to lwo or three.
 Theprocedure tomove themesh isdescribed asfollows.
Ateach time step, the instantaneous positions of the points on

+
the wing are prescribed while the points on the outer
boundary are held fixed. The displacements of the interior

points 8x,Sy,8 z are determined by solving the static

equilibrium equations which result from a summation of

i
forces at each point in the x, y, and z directions. This
solution is approximated by using a predictor-correcter
procedure which first predicts the displacements of the
interior points by a linear extrapolation of displacements
from the two previous time levels according to

gX,. j.==2 8:,.=. =-Sx,n-1j =

_'y,. =.== 2 6_ ,.i. - _,:ij. =

(3a)

(3b)
1 !-- |
,-..,.,.,.

(a) original grid (for reference).

n n-1
gz =26z -Sz (3c)
t, I. II t,J,k I,I ,la I

and then corrects the displacements using several Jacobi

iterations of the static equations written as

_ -,'. +...+ k =,jk_

_,+l '* _ j k .+., i.j.k ..... (4a)
_',a. k k +...+ k
i + _-.j, _ i.j. _-_.

k +'
_+,,.,.+...+ ki.j.
;.,., k--;
,_, ,a .... (4b)
I_ n+I __
_,.,._ k t ,
+ ...+ k
i+3.J,k i, j,k-_

n+l k +'.,., _'. + .l.l

+...+ k i, j. k--_
. _',l,J,* -I
_= -- (b) plunge upward one chordlenglh.
,.+., k + ...+ k (4C)
i*_.j.k i.j,,,-_

For the applications performed in the present study, two to

four Jacobi iterations were sufficient to accurately move the
mesh.

To demonstrate mesh movement using the deforming

mesh algorithm, consider the coarse grid about a NACA 0012
airfoil that is shown in Fig. 1 (a). The grid is of C-type
mesh topology and has 43 points in the "wrap-around"
direction and 11 points in the outward direction. It is used
only to illustrate how the mesh moves. In this example, the
airfoil was plunged for one cycle of slnusoidal motion with an
amplitude of one chordlength. The mesh at the maximum
'm
" ..--= =.,..

plunge displacement is shown in Fig. 1 {b) and the mesh at

the minimum plunge displacement is shown in Fig. 1 (¢).
The mesh moves smoothly as the airfoil plunges, and the
procedure is completely general In that it can treat realistic
airfoil or wing motions Including aeroelastic transient-type
motion.

Since the mesh can now deform to accommodate the

aeroelastically deforming wing, the flow solver in the CFL3D
code was modified to include an extra term in the time- (c) plunge downward one chordlength.
discretization of the governing equations, to account for the
mesh deformation. Specifically, the modification involves
Fig. 1 Sequence of grids about the NACA 0012 airfoil
the change in cell volume when the mesh deforms. The which illustrates how the mesh moves for a
algorithm changes are derived by first writing _/at in the plunging airfoil.
 governing equations (Eq. (1)) as a(QV)/at, where V is the where A and B are coefficient matrices that result from the
cell volume. The original flow solver assumed thal the cell change of variables x i = [qi Cli]T and u i is the nondimensional
volume does not change in time so that
generalized force weighted by mode i. Equation (7) is
integrated in time using the modified slate-transition matrix
a(o V_: va__o structural integrator 18 implemented as a predictor-
at at (5a) corrector procedure, which first uses a linear extrapolation
of ui from previous time steps as
However, if the cell volume changes in time, as it does when
n+l n n n-1
the mesh deforms, the time derivative becomes
21 =¢xi+eB(3ui-u i )/2 (Sa)

a(Q V). VaQ+QaV n+l n + 1 n+l

_" = _ "_- (Sb) to compute 21 , the prediction for x t . Then, _1 is
used to compute the flow field and evaluate the load vector
which requires the implementation of the Qav/al lerm n+l
within the CFL3D algorilhm. gi These values are then used in the corrector step to
determine xin+l given by

Pulse Transfer-Function Analysi.q

Generalized aerodynamic forces that are used in n+l n n+l n
x I =¢)xi+@Blg t +ui)/2 (8b)
aeroelaslic analyses are typically obtained by calculating
several cycles of a harmonically forced oscillation with the
determination of the forces based on the last cycle of motion. In Eqs. (8a) and (8b), • is the state-transition matrix and
This method of harmonic oscillation requires one flow 8 is the integral of the state transition matrix from time
calculation for each value of reduced frequency that is of step n to n+l.
interest. In contrast, the generalized forces may be
determined for a wide range of reduced frequency in a single Modal Identification Techniaua
flow calculation by the pulse transfer-function analysis. In Damping and frequency characteristics of the aeroelastic
the pulse analysis, the forces are computed indirectly from responses are estimated from the response curves by using
the response of the flow field due to an exponentially shaped the modal identification technique of Bennett and
pulse. The analysis assumes that the system is linear which Desmarais. 19 The modal estimates are determined by a least
is a reasonable assumption even for transonic cases, since squares curve fit of the responses of the form
experience has shown that the response for harmonic or
aeroelastic motion is, in general, locally linear lor small
amplitudes of oscillation.
qi("r) =ao+ _. e °iT
i-1 [aJ cos (_ iT) +bJ sin( (° IT)] (9)

Time-Marchina Aeroalastic Analysis i=1,2 ....

where m is the number of modes.
In this section the aeroelastic equations of motion, the
time-marching solution procedure, and the modal
identification technique are described.
Results and Discussion

Aeroelastic Eauations of Motion Results are presented in this section for the NACA 0012
The aeroelastic equations of motion that were airfoil and a 45 ° sweplback wing, computed using CFL3D, to
verify the deforming mesh capability and to assess the code
incorporated within CFL3D were derived by assuming that
for aeroelaslio analysis. The accuracy of these results is
the general motion of the wing is described by a separation of
determined by making detailed comparisons with CFL3D
time and space variables in a finite modal series. 16 This
modal series involves the summation of free vibration modes calculations performed using a rigidly moving mesh,
published results oblained using altemative computational
weighted by generalized displacements. Considering methods, and available experimental data.
Lagrange's equations leads to the equations of motion which
can be written for each mode i as
=

mtqi+ ClCll +klqi=Qi (6)

where qi is the generalized normal mode displacement, mi is

the generalized mass, c i is the generalized damping, ki is the
generalized stiffness, and Qi is the generalized force
computed by integrating the pressure weighted by the mode
shapes.

Time-Marchina Solution
The aeroelastic equations of motion are integrated in
i---- --
time in a manner similar to that described by Edwards, et
al. 17,18 The formulation is implemented herein for
multiple degrees-of-freedom or mode shapes of a wing
following Ref. 16. Each normal mode equation represented
by Eq. (6) can be expressed in state-space form as

x i = Ax i + Bu i (7) Fig. 2 Partial view of 159 x 49 C-type mesh about the

NACA 0012 airfoil.
NACA 0012 Airfoil Results 1.2
Calculations were performed for the NACA 0012 airfoil
by using the CFL3D code run in a 2-D mode. The results .8
were obtained using a 159 x 49 C-type mesh, a partial view Experiment
of which is shown in Fig. 2. The outer boundaries of the
mesh were located approximately fifteen chordlengths from .4
the airfoil and there are 110 points which lie on the airfoil _ CFL3D
surface. Calcula|ions were performed for the airfoil at M, =
!
0.8 and zero degrees angle of attack. In these calculations, -Cp o
the Euler equations were solved using the CFL3D code with a i
third-order accurate upwind-biased spatial discretization --.4
B

and flux-vector splitting. Steady results are compared with

the experimental pressure data of Ref. 20 and the unsteady
results are compared with the parallel computational results
of Ref. 11, obtained using the 2-D Euler code of Ref. 21.
-1.2 i I I I I I
_;teady Pressure Comoarisons. - The calculated steady
0 .2 .4 .6 .8 1.0
pressure distribution along the upper surface of the airfoil
is compared with the experimental data in Fig. 3. These ×/c
pressures indicate that there is a moderately strong shock
wave near the airfoil midchord, which is accurately
predicted by CFL3D in both strength and location. Such a Fig. 3 Comparison of upper surface steady pressure
good comparison suggests that viscous effects for this case distributions on the NACA 0012 airfoil at M, - 0.8
are relatively small and, thus, the flow can be modeled and % = 0°.
accurately by the Euler equations.

Generalized Force Comparisons. Generalized

aerodynamic forces for the NACA 0012 airfoil are presented Pulse - rigid mesh
in Fig. 4. The results are plotted as real and imaginary _m_ Pulse - deforming mesh
components of the unsteady forces, Aij, as 8 function of - O Harmonic - deforming mesh
reduced frequency k. Both plunge and pitch-about-the- 4
D Harmonic (Rausch)
quarter-chord motions were considered, which are defined
as modes 1 and 2, respectively. Thus, for example, A12 is 0=b
the lift coefficient due to pitching. Figure 4 compares
results obtained using several different methods including: All
-4
(1) the pulse analysis with a rigidly moving mesh; (2) the
pulse analysis with a deforming mesh; (3) harmonic motion Imaginary
with a deforming mesh; and (4) the harmonic Euler results -8
from Ref. 11. With the rigidly moving mesh, the mesh
1
simply translates for airfoil plunge motion and rotates for
airfoil pitch motion. The harmonic results were obtained at "
six values of reduced frequency: k - 0.0, 0.125, 0.25, 0.5,
0.75, and 1.0. The amplitudes of motion were 0.01
chordlengths and 0.1 degrees for plunge and pitch, A2_
respectively, in both harmonic and pulse analyses. -1

As shown in Fig. 4, the forces from the pulse analysis -2

obtained using the deforming mesh agree very well with the
forces obtained using the rigidly moving mesh. This good
agreement between the two sets of forces tends to verify the
deforming mesh capability that was implemented within
CFL3D. As further shown in Fig. 4, the pulse results agree
well with the forces from the harmonic analysis, for both I
plunge and pitch motions, for the entire range of reduced
frequency that was considered. The harmonic analysis, A12
however, is considered to be the more accurate of the two
sets of calculations, since the local linearity assumption in Imaginary
-I 0 L-
the pulse analysis is questionable for transonic cases.
Furthermore, the generalized forces determined using the
harmonic analysis agree well with the Euler forces of
Rausch, et al. 11 which gives additional confidence in the
accuracy of the deforming mesh capability that was
implemented.

Aeroelastic Comparisons. - Aeroelaslic results are

presented for a much-studied case designated as Case A of
Isogai, 22 which has normal modes similar to those of a
streamwise section near the tip of a sweptback wing. The 0 .2 .4 .6 .8 1.0
wind-off bending and torsion natural Irequencies are 71.33 k
and 535.65 rad/sec, respectively. The pivot point for the
bending mode is located 1.44 chordlenglhs upstream of the
leading edge of the airfoil. The pivot point for the torsion Fig. 4 Comparisons of generalized aerodynamic forces for
mode is 0.068 chordlengths forward of midchord. These the NACA 0012 airfoil at M, - 0.8 and a o - 0 °.
 Table 1 Comparisons between aeroelastic solutions for the NACA 0012 airfoil
at M- = 0.8 and _o = 0° for Case A.

Mode 1 Mode 2
Method o/(oa (o/(ou olmu (ol(oa

.2 CFL3D time-marching -.011 .794 -.091 5.363

Euler (Rausch) time-marching -.011 .790 -.068 5.353

.5 CFL3D time-marching .004 .914 -. 1 85 5.347

Euler (Rausch) time-marching .000 .913 -.148 5.349

.8 CFL3D time-marching .026 1.027 -. 1 73 5.270

Euler (Rausch) time-marching .017 1.022 -.223 5.317

-- CR.3D mode shapes and natural frequencies were determined by

performing a free vibration analysis with the aeroelastic
------ Euler (Rausch)
equations written in the traditional form of plunge and pitch
x 10 -=
degrees of freedom. In this analysis, the following
structural parameter values were used: a - -2.0, x_ - 1.8,
q2 rct= 1.865, (oh - 100 rad/sec, and (o_ - 100 rad/sec. Also,
__ L 0-O.2 the airfoil mass ratio was p = 60. Generalized displacements
corresponding to the bending and torsion modes are defined as
ql and q2, respectively. Initial conditions for the time-
q2 marching aeroelastic analysis were 41(0) = 2.0 and _12(0) =
-1L_V _ "_ _" 0-o.5 0.01.
3i-
Aeroelastic results for Case A were obtained for several
21 - values of nondimensional dynamic pressure including _; -
0.2, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8, 1o obtain conditions
which bracket the flutter point. Figure 5 shows time
q2 0
responses_of generalized displacement of the second coupled
__ (3- o.s mode for Q = 0.2, 0.5, and 0.6 which correspond to stable,
near neutrally stable, and unstable aeroelastic behavior,
--2--
respectively. Also plotted are the corresponding responses
_3 _ I I I I I reported in Ref. 11. A comparison of these responses
o .04 .08 .12 .16 .20 indicates that the time-marching aeroelastio results from
T the CFL3D code agree well with those from the Euler code of
Ref. 11, which tends to verify the aeroelastic modeling
procedures that were implemented. Shown tn Fig. 6 are the
Fig.5 Comparisons of generalized displacements for the
two-mode curve fits of the CFL3D responses which are
NACA 0012 airfoil at M, ,,, 0.8 and u o ,, 0 ° for
excellent approximations to the original data. The component
Case A. modes from these curve fits are shown in Fig. 7 for the three

-- Data -- Mode 1

------ Two-mode fit ----- Mode 2

x 10 "_ x 10 "3
I

q2 0 Amplitude

-1 O= 0.2 -I L O- 0.2

q2 0 , Amplitude

-I
3 3-

2 2-

q2
I
0
:AA Amplitude 0 _rr r'-'l - -

-I V (_ - 0.8

-2 --2 --

I I I I I
-3 _30 I I I I I
0 .04 .O8 .12 .16 .20 .04 .08 .12 .16 .20
T T

Effects of nondimensional dynamic pressure on the Fig. 7 Effects of nondimensional dynamic pressure on the
Fig. 6
component modes of the aeroelastio system for the
generalized displacement of the second coupled
mode for the NACA 0012 airfoil at M, - 0.8 and NACA 0012 airfoil at M, = 0.8 and u o - 0° for
Case A.
% = 0 ° for Case A.
values
of_ that were considered in Figs. 5 and 6. The results
of Fig. 7 show that the component modes consist of a
dominant mode corresponding to bending (mode 1) and a
second higher-frequency mode corresponding to torsion
_Y
(mode 2). Damping and frequency estimates from this
analysis are compared with similar values from Rausch, el
a1.11 in Table 1. These comparisons Indicate that the CFL3D Id"v_ 1, f 1 . m,60 HZ
values correlate well with the Euler values from Ref. 11.
Also, the flutter value for _ computed by quadratic
Mmll 2, f 2 • 11.17 HI
interpolation of the damping values, was 0.48 for CFL3D
which compares with 0.50 as reported in Ref. 11. Linear
theory at M, - 0.8, which of course does not include
transonic effects, predicts a much higher flutter value of
1.89.

45 ° Sweotback Win 0 Results

Calculations were performed for a simple well-defined Mo_3,13. 48,.35
l,,Iz
wing, to assess the CFL3D code for three-dimensional
aeroelastic applications. The wing that was analyzed was a
semispan wind-tunnel-wall-mounted model fhal has a Idgdl4,f4.91.,,$4Hz
quarter-chord sweep angle of 45 °, a panel aspect ratio of
1.65, and a taper ratio of 0.66.15 The wing is an AGARD Fig. 10 Oblique projections of natural vibration modes of
standard aeroelastic configuration which was tested in the 45" sweptback wing.
Transonic Dynamics Tunnel (TDT) at NASA Langley Research

Sj
Center. A planview of the wing is shown in Fig. 8. The wing
has a NACA 65A004 airfoil section and was constructed of
laminated mahogany. In order to obtain flutter for a wide
range of Mach number and density conditions in the TDT,
holes were drilled through the wing to reduce its stiffness.
To maintain the aerodynamic shape of the wing, the holes
were filled with a rigid foam plastic. A photograph of the
¢

M0dl 1, f 1 ,. 9.110 Nz Modl 2, f 2 - liE17 HI

I •

| 1

Jkde 3,13 . 41._ I'lz M_e 4, t 4 - III..M l'lZ

Fig. 11 Deflection contours of natural vibration modes of

45 ° sweptback wing.

Fig. 6 Planview of 45 ° sweptback wing.

_J_:'...

Fig. 9 45 ° sweptback wing in the NASA Langley Transonic Fig. 12 Partial view of 193 x 33 x 41 C-H-type mesh
Dynamics Tunnel. about the 45 ° sweplback wing.

6LACK Al',ii),','i-:;i£ i,-,OI-OGRAPH'

wingmounted
intheTDTis shown inFig.9. Thewingis Concludina Remarks
modeled
structurally
usingthefirst four natural vibration Modifications to the CFL3D three-dimensional unsteady
modes which are illustrated in Figs. 10 and 11. Figure 10 EulerlNavier-Stokes code for the aeroelastic analysis of
shows oblique projections of the natural modes while Fig. 11 wings were described. The modifications involve including a
shows the corresponding deflection contours. These modes deforming mesh capability which can move the mesh to
which are numbered 1 through 4 represent first bending, continuously conform to the instantaneous shape of the
first torsion, second bending, and second torsion, aeroelastlcally deforming wing, and including the structural
respectively, as determined by a finite element analysis. equations of motion for their simultaneous time-integration
The modes have natural frequencies which range from 9.6 Hz with the governing flow equations. Calculations were
for the first bending mode to 91.54 Hz for the second torsion performed using the Euler equations to verify the
mode. modifications to the code and as a first step toward
aeroelastio analysis using the Navier-Stokes equations.
Aeroelastic results were obtained for the 45 ° sweptback
wing using a 193 x 33 x 41 C-H-type mesh, a partial view Results were presented for the NACA 0012 airfoil and a
of which Is shown in Fig. 12. Calculations were performed 45 ° sweptback wing to demonstrate applications of CFL3D
for the wing at M= =0.9 and zero degrees angle of attack. for generalized force computations and aeroalastic analysis.
In these calculations, the Euler equations were solved using Detailed comparisons were made with published Euler
the CFL3D code with a second-order accurate upwind-biased results for the NACA 0012 airfoil which indicated very good
spatial discretization and flux-vector splitting. Aeroelastic agreement for generalized forces due to harmonic motion in
transients were obtained for several values of dynamic pitch or plunge, and good agreement for aeroelastic
pressure Q, to obtain conditions which bracket the flutter transients corresponding to stable, neutrally stable, and
unstable aeroelastic behavior. This favorable correlation
point. Figure 13 shows time responses of generalized
tends to verify the deforming mesh capability and the
displacement of the first bending mode for Q=0.9 Q exp, 1.0
aeroelastic modeling procedures that were implemented
Q exp, and 1.1 Q exp, where Q exp is the experimental
within CFL3D. Aeroelastic transients were obtained for a
flutter dynamic pressure value. Also shown in Fig. 13 are
the two-mode curve fits of the responses which are very 45 ° sweptback wing which also demonstrated stable,
good approximations to the original data. The component neutrally stable, and unstable behavior. The resulting
modes from these curve fits are shown in Fig. 14 for the flutter dynamic pressure, determined by interpolation of the
dominant damping values, was within 8% of the
three values of Q that were considered in Fig. 13. The
results of Fig. 14 show that Ihe component modes consist of a experimental flutter value and within 1% of a transonic
small-disturbance result.
dominant mode corresponding to first bending (mode 1) and a
higher-frequency damped mode corresponding to first
torsion (mode 2). Also, the flutter value for Q computed by
quadratic interpolation of the damping values was 0.92
Qexp. Although the calculated value is slightly low in The work constitutes a part of the first author's M.S.
comparison with the experimental value, it is within 1% of thesis at Purdue University and was supported by the NASA
the value predicted by the CAP-TSD transonic small- Langley Graduate Aeronautics Program under grant NAG-I-
372. Also, the authors would like to thank Jim Thomas and
disturbance code, 16 which tends to verify the computational
Kyle Anderson of the Analytical Methods Branch, NASA
aeroelasticity methods of the present study.
Langley Research Center, Hampton, Virginia, and Sherrio

Data Mode 1

Two-mode fit m m_ Mode 2

ql
A Amplitude
.0

v Q = 0.9 Qexp Q = 0.9 Qexp

ql Amplitude
Q = 1.0 Qexp Q = 1.0 Qexp

.02 A .02 ;_'_

ql

--o_

-.06
0

-.o,-V \ I I I I
Q ,. 1.1 Qexp

I
Amplitude

"°'
-.02 !
-.06
0 ,,.,,. -

I
O ,, 1.1 Qexp

] I
0 .O4 .08 .12 .16 .20 0 .IN .O8 .12 .16 .20
T T

Fig. 13 Effects of dynamic pressure on the generalized Fig. 14 Effects of dynamic pressure on the first two
displacement of the first bending mode for the 45 ° component modes of the aeroelastic system for the

sweptback wing at M =0.9and ,',o=0 °. 45 ° sweplbeck wing at M =0.9 and ¢Xo=0 ° .

KristofVigyan
Research
Associates,
Hampton,
Virginia,
for 12Anderson, W. K.; Thomas, J. L.; and Van Leer, B.:
many fruitful discussions during the course of the present Comparison of Finite Volume Flux Vector Splittings for the
work. Euler Equations, AIAA Journal, VoI. 24, September 1986,
pp. 1453-1460.

F]eferences 13Anderson, W. K.; Thomas, J. L.; and Rumsey, C. L.:

1Edwards, J. W.; and Thomas, J. L.: Computational Extension and Application of Flux-Vector Splitting to
Methods for Unsteady Transonic Flows, AIAA Paper No. 87- Unsteady Calculations on Dynamic Meshes, AIAA Paper No.
0107, January 1987. 87-1152, June 1987.

2Borland, C. J.; and Rizzetta, D. P.: Nonlinear Transonic 14Batina, J. T.: Unsteady Euler Algorithm With
F/utter Analysis, AIAA Journal. vol. 20, November 1982, Unslruclured Dynamic Mesh for Complex-Aircraft
Aeroelastic Analysis, AIAA Paper No. 89-1189, April 1989.
pp. 1606-1615.

3Batina, J. T.; Seidel, D. A.; Bland, S. R.; and Bennett, R. 15yates, E. C., Jr.; Land, N. S.; and Foughner, J. T., Jr.:
Measured and Calculated Subsonic and Transonic Flutter
M.: Unsteady Transonic Flow Calculations for Realistic
Aircraft Configurations, Journal of Aircraft, vol. 26, Characteristics of a 45 ° Sweptback Wing Planform in Air
and in Freon-12 in the Langley Transonic Dynamics Tunnel,
January 1989, pp. 21-28.
NASA TN D-1616, March 1963.

41sogai, K.; and Suelsuga, K.: Numerical Simulation of

16Cunningham, H. J.; Balina, J. T.; and Bennett, R. M.:
Transonic Flutter of a Supercritical Wing, National
Aerospace Laboratory, Japan, Rept. TR-276T, August 1982. Modern Wing Flutter Analysis by Computational Fluid
Dynamics Methods, Journal of Aircraft. vol. 25, October
1988, pp. 962-968.
51de, H.; and Shankar, V.J.: Unsteady Full Potential
Aeroelastic Computations for Flexible Configurations, AIAA
Paper No. 87-1238, June 1987. 17Edwards, J. W.; Bennett, R. M.; Whitlow, W., Jr.; and
Seidel, D. A.: Time-Marching Transonic Flutter Solutions
6Bendiksen, O. O.; and Kousen, K. A.: Transonic Flutter Including Angle-of-Attack Effects, _, Vol.
Analysis Using the Euler Equations, AIAA Paper No. 87- 20, November 1984, pp. 899-906.
0911, April 1987.
18Edwards, J. W.; Bennett, R. M.; Whitlow, W., Jr.; and
7Kousen, K. A.; and Bendiksen, O. O.: Nonlinear Aspects of Seidel, D. A.: Time-Marching Transonic Flutter Solutions
the Transonic Aeroelastic Stability Problem, AIAA Paper No. Including Angle-of-Attack Effects, AIAA Paper No. 82-
88-2306, April 1988. 3685, May 1982.

8Wu, J.; Kaza, K. R. V.; and Sankar, L. N.: ATechnique for 19Bennett, R. M.; and Desmarais, R. N.: Curve Fitting of
the Prediction of Airfoil Flutter Characteristics In Separated Aeroelaslic Transient Response Data with Exponential
Flow, AIAA Paper No. 87-0910, April 1987. Functions, In "Flutter Testing Techniques," NASA SP-415,
May 1975, pp. 43-58.

9Reddy, T. S. R.; Srivastava, R.; and Kaza, K. R. V.: The

2OMcDevitt, J. B.; and Okuno, A. F.: Static and Dynamic
Effects of Rotational Flow, Viscosity, Thickness, and Shape on
Pressur_ Measurements on a NACA 0012 Airfoil in the Ames
Transonic Flutter Dip Phenomena, AIAA Paper No. 88-
High Reynolds Number Facility, NASA TP-2485, June
2348, April 1988.
1985.

lOGuruswamy, G. P.: Time-Accurale Unsteady 21Batlna, J.T.: Unsteady Euler Airfoil Solutions Using
Aerodynamic and Aeroelastic Calculations of Wings Using
Unstructured Dynamic Meshes, AIAA Paper No. 89-0115,
Euler Equations, AIAA Paper No. 88-2281, April 1988. January 1989.

11Rausch, R. D.; Batina, J. T.; and Yang, T. Y.: Euler

221sogai, K.: Numerical Study of Transonic Flutter of a
Flutter Analysis of Airfoils Using Unstructured Dynamic
Two-Dimensional Airfoil, National Aerospace Laboratory,
Meshes, AIAA Paper No. 89-1384, April 1989.
Tokyo, Japan, TR-617T, July 1980.
 Report Documentation Page

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TM-102733

4. Title and Subtitle 5. Report Date

Aeroelastic Analysis of Wings Using the Euler November 1 990

Equations with a Deforming Mesh 6. Performing Organization Code

7. Authoris) 8. Performing Organization Report No.

Brian A. Robinson
John T. Batina
10. Work Unit No.
Henry T. Y. Yang

5 05 -63 -5 0-I 2
9, Performing Organization Name and Address
fl. Contract or Grant No.
NASA Langley Research Center
Hampton, Virginia 23665-5225

13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address
Technical Memorandum
National Aeronautics and Space Administration
Washington, DC 20546-0001 14. Sponsoring Agency Code

15. Suppiernentary Notes

Presented as AIAA Paper No. 90-1.032 at the AIAA/ASME/ASCE/AHS/ASC 31st Structures,

Structural Dynamics, and Materials Conference, Long Beach, California, April 2-4,
1990.
Brian A. Robinson: McDonnell Aircraft Company, St. Louis, Missouri; John T. Batina:
Langley Research Center. R_mnton. V_r_n1_" Rpnry T_ Y_ Y_na_ P,,r_,,_ 1]n_r_v
16. A_re_ _" West Lafayette, 16dian
Modifications to the CFL3D three-dlmenslonal unsteady Euler/Navler-Stokes code
for the aeroelastic analysis of wings are described. The modifications involve

including a deforming mesh capability which can move the mesh to continuously
conform to the instantaneous shape of the aeroelastlcally deforming wing, and
including the structural equations of motion for their simultaneous tlme-integration
with the governing flow equations. Calculations _re performed using the Euler
equations to verify the modifications to the code and as a flrst-step toward
aeroelastlc analysis using the Navler-Stokes equations. Results are presented for
the NACA 0012 airfoil and a 45 ° sweptback wing to demonstrate applications of CFL3D
for generalized force computations and aeroelastlc analysis. Comparisons are made
with published Euler results for the NACA 0012 airfoil and with experimental flutter
data for the 45 ° swepthack wing to assess the accuracy of the present capability.
These comparisons show good agreement and, thus, the CFL3D code may be used with
confidence for aeroelastic analysis of wings. The paper describes the modifications
that were made to the code and presents results and comparisons which assess the
capa bil it y.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Unsteady Aerodynamics
Unclassified - Unlimited
Computational Fluld Dynamic s
Transonic Flow
Subject Category 02
Aero ela st ic it y

19. Security'Clu,_. (of thin report) 20. Security Classif. (of this page) 21. No. of pages 22. Price

Unclassif led Uncla ssif ied 1 0 A02

NASA FORM 1626 OCT 86