A solution procedure was developed to investigate the two-dimensional, one- or two-dimensional flutter characteristics of arbitrary airfoils. This procedure requires a simultaneous integration in time of the solid and fluid equations of motion. The fluid equations of motion are the unsteady compressible Navier-Stokes equations, solved in a body-fitted moving coordinate system using an approximate factorization scheme. The solid equations of motion are integrated in time using an Euler implicit scheme. Flutter is said to occur if small disturbances imposed on the airfoil attitude lead to divergent oscillatory motions at subsequent times. The flutter characteristics of airfoils in subsonic speed at high angles of attack and airfoils in high subsonic and transonic speeds at low angles of attack are investigated. The stall flutter characteristics are also predicted using the same procedure

Sankar, L. N.

Srivastava, R.

Wu, J. C.

English

NASA Technical Reports Server

N88-23249
APPLICATION OF NAVIER-STOKES ANALYSIS TO STALL FLUTTER*
J.C. Wu, R. Srivastava, and L.N. Sankar
Georgia Institute of Technology
Atlanta, GA
ABSTRACT
A solution procedure has been developed to investigate the two-dimensional,
one- or two-degree-of-freedom flutter characteristics of arbitrary airfoils.
This procedure requires a simultaneous integration in time of the solid and
fluid equations of motion. The fluid equations of motion are the unsteady com-
pressible Navier-Stokes equations, solved in a body-fitted, moving coordinate
system using an approximate factorization scheme. The solid equations of
motion are integrated in time using an Euler implicit scheme. Flutter is said
to occur if small disturbances imposed on the airfoil attitude lead to diver-
gent oscillatory motions at subsequent times.
The flutter characteristics of airfoils in subsonic speed at high angles of
attack and airfoils in high subsonic and transonic speeds at low angles of
attack are investigated. The stall flutter characteristics were also predicted
using the same procedure. Results of a number of cases are included and com-
pared with numerical and experimental data where available. The effects of
mass ratio, initial perturbation, mean angle of attack, viscosity, and shape
and thickness, on the flutter boundary are also investigated.
*This work was performed under NASA Grant NAG 3-730. The authors acknowl-
edge Drs. K.R.V. Kaza and T.S.R. Reddy for the technical discussions and their
suggestions.
1-309
https://ntrs.nasa.gov/search.jsp?R=19880013865 2020-03-20T06:21:34+00:00Z
GOVERNINGEQUATIONS
The fluid equations of motion used in the present formulation are the compres-
sible Navier-Stokes equations. These equations are written below in a conserv-
ative form. These are solved in a body-fitted, moving coordinate system using
an appropriate factorization scheme.
_tq + 5zE + 6yF = Re-I(_xR + 6yS)
WHERE
q_
1 F=r v1 R=Fo1pu pu2 + p / puv | TxxL.uvi l.v,÷Pl i'.y,
L,,(e+ P)J Lv(e+ P)j LR4j
S _
STRUCTURAL MODEL FOR 2-DOF SYSTEM
0
7xy
7"yy
S4
C0--M-32610
The structural dynamic model considered is a 2-DOF (pitching and plunging
motion) system. An Euler implicit scheme was used to integrate the structural
governing equation. The fluid and the solid equations were simultaneously
integrated in time to monitor how lift, moment, and drag vary with time.
_y
----_ L!
ELASTICAXISJ "ibah -
b _
_L
$S CENTER
X ii
C0-•_32611
GOVERNING EQUATION OF THE 2-OOF STRUCTURAL MODEL
I_ + Sh"+ g_,-I- K_o_= M(t)
mh' + S_ + ghh+ Khh = -L(t)
• EULER IMPLICIT SCHEME FOR I1, h', etc.
hTM_h n hn+l_2h n+h n-1
_--_, h'=
A t At 2
n LEVELI CL(t)' CM(t)
AERODYNAMIC ISOLVER
n+l LEVELJr _(t), _(t),h(t), h(t)
' iI STRUCTURAL DYNAMIC[SOLVER
1-310
COMPARISONSOFUNSTEADYAIRLOADSONA NACA0012
AIRFOIL EXPERIENCINGDYNAMICSTALL
The present Navier-Stokes solver is able to obtain time-accurate results in
highly separated flows. The lift, drag, and momenthysteresis loops are shown
here and comparedwith experiments by McAlister et al (1982). The case is
shownof a NACA0012 airfoil oscillating in pitch with the meanangle of oscil-
lation 15 degrees and I0 degrees of amplitude of oscillation. The solver cor-
rectly predicts (i) the near-linear increase in lift during the upstroke;
(2) the dynamic stall which causes rapid variations in lift, drag, and moment
alike; and (3) the post stall recovery phase of the flow during the downstroke.
DRAGCOEFF.
MOMENT COEFF.
1.2 --
0.6
0.0
M==0.283, Re=3.45x106, kb=0.151
PRESENT
EXPERIMENT
-- /
o.1
0.0 ____ ___,..._
-0.1 --
-0.2
-0.3
-0.4 -- _/
_o.55. I I [ [10.0 15.0 20.0 25.0
ANGLE OF ATTACK, o_
2.5 --
2.0
1.5
LIFT COEFF.
1.0
0.5
o.o :'__ I I
5.0 10.0 15.0 20.0 25.0
CD-88-32613
1-311
VARIATIONOF LIFT ANDPITCHINGMOMENTCOEFFICIENT
FORPLUNGINGMOTION
The present code's ability to handle unsteady, transonic flows in a time-
accurate manner is illustrated in the figure below. The case is shownof a
NACA64A010airfoil oscillating sinusoidally in plunge at a free stream Mach
numberof 0.8 at zero meanangle of attack. The lift and pitching momenthis-
tory are plotted as a function of phase, and are comparedwith the Euler calcu-
lations performed by Steger (1978). Very good agreement is observed between
the two solvers.
NACA64A010 AIRFOIL
h = - M= sin (1°) sin ((_t)
M,,, = 0.8, or=O, kb= 0.2
PRESENT(EULER)
STEGER(EULER)
o,°r-/_
oo_r/ \
ooo_
c, -oo_- _ ,_
::l,O,r,,,
0 60 120 180 240 300 360
CM
0.010
0.005
0.000
-0.005
-0.010
-0.015
0 60 120 180 240 300 360
PLUNGEANGLE, DEG
CD-88-32614
1-312
EFFECTSOFAIRFOIL-AIRMASSRATIOONFLUTTERSPEED
The present technique for the prediction of stall flutter was validated for
transonic flutter calculations where reliable numerical solutions exist. The
airfoil is a NACA64A006airfoil at a free stream Machnumber 0.85, and the
flow was assumedto be inviscid. The flutter speed predicted by the present
theory is plotted as a function of the airfoil-alr mass ratio (Wuet al.,
1987). For comparison, the results from the LTRAN2and UTRANS2(Ballahus,
1978) and UTRANS2 (Farret al., 1974) codes are shown. It is seen that the
Euler results agree very well with the prediction of the UTRANS2 code, while
only a qualitative agreement between the present results and the LTRAN2 code
could be found.
FLUTTER
SPEED
U/b_.
10
2
0
------ UTRANS2
I-I PRESENT(EULER)
-- LTRAN2
NACA64A006 AIRFOIL
ah = - 0.5
X,_= 0.25
r(_= 0.5
_h/_,_ = 0.2
M_ = 0.85
I I I
100 200 300
AIRFOIL-AIR MASS RATIO, #
CD-88-32615
1-313
RESPONSESOFTHE2-DOFSOLID-FLUIDSYSTEMAS A
FUNCTIONOFTIME
The following figure illustrates the time responses of a 2-DOFflutter calcula-
tion. The flow is assumedto be inviscid at free stream Machnumber 0.85. A
NACA64A006airfoil was released after the forced sinusoidal oscillation and
was allowed to follow pitching and plunging motions dictated by the structural
dynamic equations. By parametrically varying the airfoil air mass ratio during
this phase of the calculations, it led to dampedoscillations, neutral oscilla-
tions, stable oscillations, or divergent (flutter) oscillations.
1.00j-- 0.10
0.5 _. . 0.05
<x(DEGREE) o.oo h/b 0.00
-o.5_- I I I I -o.o5
-1.oo I -olol [ I I I I
0.20 -- __
0.10
CM x 10 0.00
-0.10
-0.20
1.00
# = 174.75 (NEUTRAL)
# = 200 (STABLE)
# = 150 (UNSTABLE)
I I I I I
2 4 6 8 10
TIME (oot) IN x RADIANS
CL
0.10
0.05
0.00
-0.05
-0.10
-0.15
0.15 -- NACA64A006 AIRFOIL
_ Moo= 0.85 ah= - 0.5
oohlu_ = 0.2
_ r_ = 0.5 x_ = 0.25
m
I I I I I
0.00 2 4 6 8 10
TIME (ut) IN x RADIANS
CD-88-32616
1-314
EFFECTSOFVISCOSITYONTHETIME RESPONSE
The effect of flow viscosity on the flutter characteristics for a 2-DOFsystem
was studied using the present solver operating in the Navier-Stokes mode. The
airfoil is a NACA64A006airfoil at a free stream Machnumber 0.85. Viscous
solution corresponds to a Reynolds numberof 9x106. The flutter boundaries
predicted by the viscous and inviscid calculations were within 2 percent of
each other, which means that in high Reynolds number transonic flutter studies,
inviscid calculations would suffice.
1.00 0.10
0.5
o_(DEGREE) 0.00
-0.5
-1.00 I I I I
CM ×10
0.20 --
0.10
0.00
-0.10
-0.20
0.00 2
hlb
INVISCID
/_= 170.0 (NEUTRAL)
VISCOUS
, ,,,, -,174"75(NEUTRAL)cL
I I I I I
4 6 8 10
TIME (o_t) IN _r RADIANS
0.05
0.00
- 0.05
-0.10 I I ] I
0.15 -- NACA64A006 AIRFOIL
0.10
0.05
0.00
-0.05
-0.10 --
-o15 I I I I I
0.00 2 4 6 8 10
TIME (o_t) IN _" RADIANS
M= = 0.85 Re= 9 x 106
O_h/_ = 0.2 ah= - 0.5 x_ = 0.25
_ V* = 5.5 kb= 0.5 r_ = 0.5
J"%%.
CD-88-32617
1-315
TIMERESPONSESOFA 2-DOF SYSTEM
EXPERIENCING STALL FLUTTER
The stall flutter calculations are carried out using the Navier-Stokes/struc-
rural dynamics solver. The case considered was a NACA 0012 airfoil, initially
subjected to a sinusoidal pitching oscillation between 5 and 25 degrees. Dur-
ing the downstroke, around 23.8 degrees, the airfoil was released and was
allowed to follow a pitching and plunging motion dictated by the structural
dynamic equations. Two dimensionless speeds V*, 4 and 8, were considered.
At the lower speed, the airfoil began to undergo a damped sinusoidal oscilla-
tion and reached a stable condition eventually. The time history for the
speed V* equal to 8, however, showed a rapidly growing oscillating motion
indicative of dynamic stall flutter.
(DEGREE)
CM
STABLEV* = 4
UNSTABLEV* = 8
INITIALANGLEOF ATTACK,23.8°
30.00 --
15.00 _0.00
_1 oo /
-30.00
ti ,,
,y\
I I I ]
0.30 --
0.15 --
ooo' ,,,,
-0.15 /i--_/i'_ _
-0.30 I
0.00 1.25 2.50 3.75 5.00
TIME ((_t) IN x RADIANS
NACA0012 AIRFOIL
M= = 0.283 Re = 3.45 x 106
Chlu,, = 0.2 ah= - 0.5
ra = 0.5 /_ = 100
2.80 --
1.40
h/b 0.00
- 1.40
-2.80 I I I 1
CL
2.40 --
1.20- I_ //
W",,/t17/\,,'"
ooo,/\tl/'_' \
-,._o_i t,.W' L,
_2.4O;.ooI I I I1.25 2.50 3.75 5.00
TIME (_ot) IN _" RADIANS
x_ = 0.25
CD-88-32618
1-316
COMPARISONFCALCULATEDFLUTTERBOUNDARIES
A comparison of flutter boundaries with the boundaries obtained by various
codes presented by Bendikson et al. (1987) is shownbelow. Twoartificial vis-
cosity models were used in the present calculations (Reddy et al., 1988).
The artificial viscosity in the present code is based on pressure gradient. In
model i, the pressure gradient was scaled by a constant coefficient, whereas
in model 2, it was scaled by the spectral radius.
The rotational effects of the flow behind the shock wave have strong effect on
the transonic flutter speed, depending on the chordwise location of the shock.
Neglecting the flow rotation effects results in predicting a higher flutter
speed.
SPEED
INDEX,
Vf/b_/'-#_
4.0 B
2.0
0.0
0.7
NACA64A010AIRFOIL
• PRESENTEULER(MODEL1)
.... PRESENTEULER(MODEL2)
• BENDIKSON'SEULERCODE _,,
A LTRAN2 _,,,,_
O HYTRAN2
_7 OPTRAN2 _7 A
_h/_ot=l.O _ ?
#=60 I ]
ah= - 2.0 x_ = 1.8 _ I
ro=,., \
I I I
0.80 0.90
MACH NUMBER, M
1.00
CD-88-32619
1-317
EFFECTOFMEANANGLEOFATTACK,INITIAL PERTURBATION,
ANDVISCOSITYONTHEFLUTTERBOUNDARY
The effect of initial perturbation, meanangle of attack, and viscosity are
shownin the figure below.
The effects of initial conditions, meanangle of attack, and viscosity on the
minima of the transonic dip seemnegligible. However, they have a significant
effect away from the dip. Similar results for meanangle of attack were
obtained by Edwardset al. (1983).
SPEED
INDEX,
Vf/b_'_
4.0 m
2.0
0"00.7
NACA64A010 AIRFOIL
0.1 ° INITIAL PERT.
A 4.0° INITIAL PERT.
2° MEAN ANGLE
O Re = 12 x 106
Wh/W(x= 1.0
ah= - 2.0 x. = 1.8
I
I
I
I
# = 60 ro_= 1.865 I
I
I I
0.80 0.90
MACH NUMBER, M
I
1.00
CD-88-32620
1-318
EFFECTOFAIRFOIL SHAPE AND THICKNESS ON THE
FLUTTER BOUNDARY
The blade thickness and shape dictate the location and strength of shock,
thereby affecting the flutter boundary. The transonic dip shifts to higher
Mach numbers for symmetric airfoils with decreasing airfoil thickness to chord
ratios. For very thin cambered airfoils, the transonic dip occurs at lower
Mach numbers.
This effect is shown in this figure.
30
25
FLUTTER 2.0
SPEED
INDEX,
VF/(b_o_V'--#_) 1.5
1.0
.5
.70
Chl_= = 1.0 ah= - 2.0 X== 1.8
p.= 60.0 r= = 1.865
- O
1
I [ -'h_"_ 't_" I
.75 .80 .85 .90 .95
MACH NUMBER,M
LINEARSUBSONICTHEORY
NACA16-010 AIRFOIL
NACA16-004 AIRFOIL
NACA16-(1.3)(04) AIRFOIL
NASA16-(1.3)(2.6) AIRFOIL
(PROPFANAIRFOIL)
CD-88-32621
1-319
FLUTTERBOUNDARYFORA SIMULATEDSR5TYPICAL
SECTIONSTRUCTURALMODEL
This figure shows the predicted flutter boundary of a simulated typical section
model of an SR5 propfan blade. The flutter Mach number predicted by the pres-
ent code is about _.5 percent lower than that predicted by linear theory and
experiment. This difference could be attributed to the simplified aeroelastic
model used in the present analysis.
1.0
FLUTTER
SPEED
INDEX, .S
VF/(b_a_/"_ )
0
.7
mm_ LINEARSUBSONICTHEORY
PRESENTCODE
0 EXPERIMENT
_h/_a m0.03 ah" -- 1.0 X== 9.84
m 115. ram1.078
I I
.8 .9
MACH NUMBER, M
REFERENCES
i
1.oo
Ballahus, W.F., and Goorjian, P.M., 1987, "Computation of Unsteady Transonic
Flow by the Implicit Method," AIAA Journal, Vol. 16, No. 2, pp. 117-124.
Bendikson, 0.0., and Kousen, K.A., 1987, "Transonic Flutter Analysis Using the
Euler Equations," AIAA Paper No. 87-0911-CP.
Edwards, J.W., Bennett, R.M., Whitlow, W., Jr., and Seidel, D.A., 1983, "Time-
Marching Transonic Flutter Solutions Including Angle-of-Attack Effects,"
Journal of Aircraft, Vol. 20, pp. 899-906.
Farr, J.L., Traci, R.M., and Albano, E.D., 1974, "Computer Programs for Calcu-
lating Small Disturbance Transonic Flows About Oscillating Airfoils,"
AFFDL-TR-74-135.
Reddy, T.S.R., Rivastava, R., and Kaza, K.R.V., 1988, "The Effects of
Rotational Flow, Viscosity, Thickness, and Shape on Transonic Flutter Dip
Phenomena," AIAA/ASME/ASCE/AHS 29th Structures, Structural Dynamics and
Materials Conference.
Steger, J.L., 1978, "Implicit Finite Difference Simulation of Flow About Arbi-
trary Geometries," AIAA Journal, Vol. 16, No. 7.
Wu, J.C., Kaza, K.R.V., and Sankar, L. N., 1987, "A Technique for the Predic-
tion of Airfoil Flutter Characteristics in Separated Flow," AIAA Paper
No. 87-0910-CP.
1-320


Application of Navier-Stokes analysis to stall flutter

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