Prediction of Unsteady Aerodynamic Coefficients at High Angles of Attack

14 Pages·2001·0.81 MB·English

by  Pamadi, Bandu N.

AIAA 2001-4077
Prediction of Unsteady Aerodynamic
Coefficients at High Angles of Attack
Bandu N. Pamadi, Patrick C. Murphy,
Vladislav Klein and Jay M. Brandon
NASA Langley Research Center
Hampton, VA 23681
AIAA Atmospheric Flight Mechanics
Conference & Exhibit
August 6-9, 2001
Montreal, Canada
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AIAA 2001-4077
PREDICTION OF UNSTEADY AERODYNAMIC COEFFICIENTS
AT HIGH ANGLES OF ATTACK
Bandu NI Pamadii* Patrick C. Murphy, +Vladislav Klein, _and Jay M. Brandon _
NASA Langley Research Center, Hampton, VA-23681.
Abstract CL Lift coefficient
rad
The nonlinear indicial response method is used to CL,_ Lift-Curve-Slope, _, per
model the unsteady aerodynamic coefficients in the low
speed longitudinal oscillatory wind tunnel test data of 3C,
the 0.1 scale model of the F-16XL aircraft. Exponential
C,,,I Rotary derivative in pitch, _:"-_-"_V)
functions are used to approximate the deficiency func-
tion in the indicial response. Using one set of oscilla- C,,=G or c.,. per rad
tory wind tunnel data and parameter identification
method, the unknown parameters in the exponential Pitching moment coefficient
functions are estimated. The genetic algorithm is used
as a least square minimizing algorithm. The assumed Slope of pitching-moment-coefficient
model structures and parameter estimates are validated de,,,
by comparing the predictions with other sets of avail- curve, --d_-_"per rad
able oscillatory wind tunnel test data.
Acceleration derivative in pitch,
Cm_
Nomenclature
aco,
_(d_" ] 'per rad
a(OO,al*(OO,a2"(_) Indicial parameters inModel I and
_2vj
Model II
F2,F3,G Deficiency functions in Model I and II
b_(eO,b_'(eO,b2"(_) Indicial parameters (exponents) in
Model I and II, l/see
J Cost function as defined in Eq. (23)
Constants inthe algebraic model 0_
CI,C2_C3 k Reduced frequency, --
for CLq 2V
M Mach number
?" Mean aerodynamic chord (mac), m
q Pitch rate, rad/sec
da,d2,d3 Constants in the algebraic model
for Cmq
t Time, sec
r,, Period of oscillation, sec
"Senior Aerospace Engineer, Vehicle Analysis Branch, Aerospace
System Concepts and Analysis Competency, Associate Fellow, AIAA.
"Senior Aerospace Engineer, Dynamics and Control Branch, Airborne V Velocity, m/sec
Systems Competency, Senior Member, AIAA.
*Emeritus Professor, Joint Institute for Advancement of Flight
Sciences, George Washington University, Associate Fellow, AIAA. X,'eJ Distance to moment reference point (in
*Senior Aerospace Engineer, Vehicle Dynamics Branch, Airborne terms of mac) from nose, m
Systems Competency, Senior Member, AIAA.
Copyright © 2001 by the American Institute of Aeronautics and Distance to center of the planform area
Astronautics, Inc. No copyright is asserted in the United States under Xcg
(in terms of mac) from nose, m
Title 17, U.S. Code. The U.S. Government has a rnyalty-free license
to exercise all rights under the copyright claimed herein for
Governmental Purposes. All other rights are reserved by the
Angle of attack, deg
copyright owner.
o) Frequenocfyoscillatormyotionin boundary condition, say angle of attack, that is applied
pitchr,ad/sec instantaneously and held constant thereafter. The indi-
cial response is a mathematical concept and as such is
Dummvyariablfeortime difficult to determine by an experiment. In general, the
indicial response can be assumed to consist of two dis-
Acronym tinct components, the noncirculatory loading and the
circulatory loading. The noncirculatory loading arises
GA Genetic Algorithm due to the sudden change in the surface boundary con-
dition such as a sudden change in the angle of attack. It
Introduction assumes a large peak value (theoretically infinity in
incompressible flow) initially but subsequently decays
With the advent of innovative high alpha control rapidly. At present, results based on the linear piston
devices such as three-axes thrust vectoring and fore- theory are used to define the initial value of the noncir-
body vortex controllers, it has become increasingly culatory loading. For two-dimensional airfoils in in-
attractive for fighter aircraft to exploit the full potential compressible flow, Lomax 4 has presented analytical
offered by flight at high angles of attack (at or beyond expressions for the early part of the decay of the
stall) to achieve air superiority in close-air combat. noncirculatory loading. For arbitrary configurations,
Since aircraft development programs tend to rely heav- such information is not available. The circulatory load-
ily on simulations to reduce the risk and cost of flight ing is initially zero and builds up gradually to its final
test programs, accurate modeling of such motions as- steady state value that can be obtained in a relatively
sumes critical importance? The traditional aerodynamic easier fashion either by CFD methods or static wind
approach based on time invariant stability and control tunnel tests. The difficulty lies in modeling thc decay of
derivatives proves to be inadequate for predicting such noncirculatory loading and the build up of the circula-
aircraft motions because it does not account for un- tory loading. Approximating the decay of the noncir-
steady aerodynamic effects that are significant at such culatory loading and the build up of circulatory loading
flight conditions. The problem of modeling unsteady with exponential functions and using linear indicial
aerodynamic effects at high angles of attack is a formi- response method, Beddoes, _and, Leishman _ have ob-
dable task because the flow field is characterized by tained satisfactory results for the oscillatory motion of
extensive flow separations, vortices, vortex interactions two-dimensional airfoils at low angles of attack in sub-
and possible vortex bursts over the wing and tail sur- sonic flow. In the linear indicial response method, the
faces. In view of this, generalized unsteady aerody- indicial response is independent of angle of attack. In
namic models that are suitable for high angles of attack other words, if we know the indiciat response at one
flight simulation are notyet available. angle of attack, the same is applicable for all other an-
gles of attack as long as aerodynamic coefficients are
NASA Langley Research Center has conducted linear with angle of attack.
low speed static and forced oscillation tests on the 0.I
scalc model of the F-16XL aircraft at high angles of In this paper, the nonlinear indicial response
attack in the Langley 12-Foot Low Speed Wind Tunnel. method developed by Tobak 23has been used to analyze
Using these experimental data and the indicial response the low speed, longitudinal oscillatory wind tunnel data
method, 2"aan attempt has been made in this paper to of the F-16XL aircraft model at high angles of attack.
develop aerodynamic models suitable for predicting The linear indicial method is not suitable for high an-
unsteady aerodynamic coefficients in the longitudinal gles of attack because, in general, the aerodynamic co-
forced oscillatory motion of the F-16XL aircraft model efficients vary nonlinearly with angle of attack. In the
at high angles of attack. nonlinear indicial response method, the indicial re-
sponse depends on angle of attack. In Tobak's 2'3 for-
The indicial response method provides a funda- mulation, the time varying part of the indicial response
mental approach to the problem of modeling unsteady is represented by the "deficiency" function. Thus, the
aerodynamic effects. One significant advantage of this task now becomes one of modeling the time history of
approach is that when the indicial response to a par- the deficiency function. In this paper, exponential func-
ticular forcing mode is known, e.g., due to angle of tions are used to approximate the time history of defi-
attack, the cumulative response to an arbitrary forcing ciency function. First, we use a model with two term
mode can be obtained using the superposition principle exponential functions, one to approximate the decay of
in the form of Duhamel integral. By definition, an indi- noncirculatory part and the other to approximate the
cial function is the response to a change in the surface decay of circulatory part. This two term exponential
2
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representatiiosncalledmodeIlandis physically con- qF
C,.(t) = C,,,(_, _(t ),q = O)+ [-_-)C,,,q
sistent because it explicitly models the decay of both
(2)
noncirculatory and circulatory components of the defi-
ciency function. According to Tobak, 23 the initial mag- -j _(t- z,"o_(_),q = O)(x(z)dz
nitudes of the noncirculatory and circulatory compo- o
nents are fixed by the theory. Therefore, the task is to
do(
model the decay of noncirculatory and circulatory com- where oc=--. In this formulation, the terms like
dt
ponents of the decay function.
and c_q are ignored since they will be of second order
in frequency. The first term on the right hand sidc of
Secondly, we use one term exponential function to
Eqs (1) is the lift coefficient in steady flow conditions
approximate the complete deficiency function as pro-
(t --+ oo)and at an angle of attack equal to o¢(t) and zero
posed by Klein 7xv et al which is called Model II in this
pitch rate. Similarly, the first term on the right hand
study. This one term model has a simple structure and
is computationally efficient because it can bc expressed side of Eq.(2) is the pitching moment coefficient undcr
in the form of an ordinary differential equation as op- the steady flow conditions. Both these coefficients can
be obtained either from static wind tunnel tests or CFD
posed to the convolution integral form of Model I. By
methods. The second terms on the right hand sides of
comparing the predictions of Model II with those of
Model I, it is proposed to evaluate the validity of Model Eq. (!) and (2) contain the familiar damping in pitch
II for predicting the longitudinal oscillatory unsteady derivatives. Usually, the isolated value of Cm or C,,q is
not known because the small amplitude forced oscilla-
aerodynamic coefficients of the F-16XL aircraft model
tion wind tunnel tests that are normally used to obtain
at high angles of attack.
pitch damping derivatives give the combined value of
In this paper, parameter identification is used to (Chr +CL_ ) or (C,,,q+C,,_). In such tests with q =oc, it
determine the unknown parameters in the exponential is not possible isolate values of CLq or Co,q.In view of
functions of Model I and Model II. Some adjustments this, both Ct.qand C,,,j are assumed unknown. The third
have been introduced in Model I to account for uncer- terms on the right hand sides of Eq.(I) and (2) with
tainties in the use of the static wind tunnel data that is convolution integrals represent the unsteady lift and
needed in Model I and to account for deviations from pitching moment coefficient respectively and contain
linear piston theory due to nonlinearities at high angles the unknown deficiency functions F2 and F3which de-
of attack. Only one set of longitudinal oscillatory test pend on angle of attack and the elapsed time t- z. Ac-
data is used for parameter estimation purposes. The cording to Tobak, 2_
other sets of available oscillatory test data are used to
validate the parameter estimates. The genetic algorithm (_, t) = C_ (c¢t = _)- CL.(c¢t) (3)
is used to minimize the sum of squares of error between
the estimated and measured time histories of lift or F3(oc.t) = C.,_(oc,t = ,.o)- C.,.(o., t) (4)
pitching moment coefficients.
where Ct_(oqt) and C,,,_(oc,t) are the nonlinear, time de-
Analysis
pendent indicial responses, and CL_(_,t = oo) and
C,,_(_,t = oo) arc their steady state values. Therefore, as
Consider the problem of estimation of unsteady lift
(t --4_), F2(oc,t) = F3(o_,t)= 0.
and pitching moment coefficients. According to the
nonlinear indiciai function formulation of Tobak, -'_ the
Let
time varying lift and picthing moment coefficients
during oscillatory motion in pitch, for first order accu-
(5)
racy in frequency, can be expressed (in Tobak's 2"3nota- y|(t) = CL(t)-C L(oo,_(t),O)-(q-_lCL, t
tion) as
(¢)
(6)
Q(t) = CL(c_,O¢(t), q= O)+ Ctq y_(t) = C.,(t)--Cm(°°,C_(t),O)-- T C.,,,
(1)
! so that
-5 _ (t - x;Cz('t),q = O)(_('Odx
(I
!
y,(t) = -_ F2(t - _;oc('t), q= O)6c(z)dz (7)
0
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American Institute of Aeronautics and Astronautics
-i like surfaces at zero angle of attack. Therefore, for ap-
y2(t) = _(t-x;c_(_),q=O)&(z)d_ (8)
plication to complex aircraft configurations such as F-
16XL operating at high angles of attack, it is necessary
We observe that y_(t) and y2(t) denote the unsteady to derive a suitable initial value for thc nonciculatory
components of time varying lift and pitching moment component. This, however was not attempted in this
coefficient respectively. study. Instead, to account for the deviations from linear
piston theory, an unknown empirical multiplier a:*(cx) is
Model I introduced inthe second term. With these assumptions,
We assume that deficiency functions F:((x,t) and _(oqt) = aj*(oOCta(oQe -r_,_
F3(ct,t) consist of two term exponential functions as (11)
given by
-a2*(OO(4)e -'q'_''
(9)
F.,(_,t)=C La (oO)e -_v_' -- I--_) e-t_(ct)t F,(oc,t) =a_*(cx)C,,,_(cx-)_e,2_,,_,
(12)
_(o_, t) = - . . -t,",i_,
(-me ttOOe
(10)
w,,>,,
_MJ "4 "'y" Substituting in Eqs. (7) and (8), wc obtain,
Here, CL,_(ct),@'_)andC,,,,(a),(-_](.¥,._j-Y,.g) _[!a,'(off'O)C_(_(_))e-'_(_(_))°-_'6_(_)d'c]
(13)
are the terms fixed by the theory, 2'3bt(cx), b,'(cx), b:(o0
and b./gx) are unknown indicial parameters which are
k \MJ, " J
assumed to be functions of angle attack, 2,_j is the dis-
tance (in terms of mean aerodynamic chord, mac) from
the nose to the moment reference point, and _ is the
distance (in terms of mac) from the nose to the center of y; (t) = -
the planform area of the vehicle. The determination of _ .4(.-/' 2,a) a2.(ot(_))e_t4,_,_,,(,__)(x(_)d z
all unknown parameters will be discussed later. The
first term on the right hand side of Eqs. (9) or (10) ap- (14)
proximates the decay of circulatory component. To
evaluate this term, we need to know the "theoretically" Note that when the "theoretically correct" value of
correct value of static lift-curve-slope, C_(cx) or the Ca(fx) and C,,,(ct) are known, a_*(ct) = 1, and for flat
pitching-moment-cu_'e slope C,,,_(o0. However, to ac- plate like surface operating at low angles of attack,
count for uncertainties in determining a "theoretically" a:*(o0 = I.
correct value of Cry(a) or C,,,,_(e0 using static wind tun-
nel tests, we introduce a multiplier at*(¢x). This un- The rotary derivatives are assumed as follows: _
known parameter is assumed to be a function of angle
of attack (with suitable upper and lower bounds). The C_q= q, o_< 20 deg (I5)
second term in Eq. (9) approximates the decay of
noncirculatory component with an initial value equal to
(cx-20"]+c (CX-20]"
C_, c, +c2_" 57.3 ) _ 57.3 ) (16)
{4). This initial value is based on the linear piston
20 deg < o_<_70 deg
theory and is supposed to be independent of planform
shape of the body. The noncirculatory component is
assumed to be uniformly distributed over the entire
planform surface of the body so that the resultant force -\57.3J \57.3) (17)
acts at the center of the planform area giving a moment
0 _<o_< 70 deg
arm (._,j - .7,) as used in Eq.(10). However, as noted
by Tobak, _nthis result is applicable mainly for flat plate where c_, c2,c3, d_,d2, and d3are unknown constants.
4
American Institute of Aeronautics and Astronautics
Model
II The model structures for C;,_ and C,,,,t as given in
Eqs. (15) to(I7) for Model Iwere also used for Model II.
We approximate the complete deficiency function
by a single exponential term vs'9as follows: Determination of Unknown Parameters
G(cc,t) =a(cz)eh'' (I 8) In this study, one set of experimental data was used
to estimate the unknown parameters in Model I and
where F,_(cc,t)denotes the deficiency function for the Model II and other sets of available experimental data
unsteady lift or pitching moment coefficient. With this were used to assess the quality of predictions made us-
assumption, the unsteady lift or pitching moment coef- ing the estimated parameters. The genetic algorithm
ficient is given by (GA) was used as a parameter identification algorithm
because GA provides a convenient environment to es-
t timate the unknown parameters when their functional
y(t) =-J a(ec('O)e-""-"6(z)d'c (I9)
dependence (model structure) is not known a priori. A
{I
brief description of the GA isgiven in the following:
Here, a(o0 is an unknown parameter that is a function
of angle of attack and b_ is an unknown constant. Note The genetic algorithms are non derivative search
that the parameters a(cz) and b_ will have different val- procedures based on models of processes in natural
ues for lift and pitching moment deficiency functions. genetics. At each iteration, the search takes place by
The time derivative ofy(t) is given by sampling a coding parameter set from a random distri-
bution, and applying the parameter iterates to a popula-
(20) tion of function evaluations. The performance of the
+(t)=-d[!a(_('O)e-_t'-+_("c)dx I
population elements is used to modify the probability
distribution for sampling in the next iteration. Algo-
rithms of this class are known to be very robust and not
=-[! (-b_)a(R(_))e-t"¢'-'_&(z)dI+a(offt))(x(t)]
particularly vulnerable to termination at local minima.
More information on GA may be obtained in Refs 11
or, and 12.
S,(t)= -bo,(t )- a(offt))&(t) (21)
The procedure used to determine the unknown pa-
rameters is as follows: First, subdivide the angle of at-
Thus, Eq. (21) is an equivalent ordinary differential
tack range of interest into a convenient number of seg-
equation for the convolution integral form (Eq.(19)) of
ments or nodes, say n, and use a linear interpolation to
the unsteady lift coefficient y(t). Given the initial value
determine the parameter value in between the nodal
y(0), this differential equation is much more convenient
points. In GA environment, the sub intervals need not
to solve compared to the evaluation of the convolution
be equal. The next step is to specify suitable values for
integral in Eq.(19).
upper and lower limits at each nodal point for a_'(_),
a:'(_), bffot), b2(ot), b((a) and b_'(ot), for Model I and
If we assume b_to be a function of angle of attack,
a(ot) and b_(ot) for Model II and upper and lower limits
the convolution integral for y(t) which still has the ordi-
for other unknown parameters such as c_, c._, c3, d_, d2,
nary differential equation of the form given in Eq.(21)
d3 for both models. The GA generates a population
is given by the following expression: *
whose elements are assigned randomly generated val-
ues for each of the unknown parameters that lie within
the specified upper and lower limits. Thus, having as-
y(t) = -_a({x('c))e _ 6_(z)dz (22)
signed values to all unknown parameters in Model I or
0
Model II, the time histories of CL(t) or C,,(t) are gener-
We note that the convolution integral in Eq.(22) is more ated for each population element. Let the estimated
complex than the convolution integral in Eq.(14) but we time history be denoted by C'L(t)or C,,,(t). Then define,
need not evaluate it. Instead, we can evaluate y(t) more
easily by solving the ordinary differential equation as [ ]+
J = _ C_(t,)-Co(t,) (23)
given in Eq. (21) with b_ replaced by b_(ot). In this
i=1
study, the functional forms for these two parameters are
not prescribed a priori. Instead, such functional forms
are determined during the estimation process.
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where,
C,,= CL or C,,, and, t_= 1.... N correspond to the
values of time at which the oscillatory data on CL(t) or
C,,,(t) is available. The problem is to estimate the nodal
values of the unknown parameters in Model I or Model
II so that the sum of the squares of the error J is mini-
!i0
mized. The GA was used as a least square minimization
algorithm with J as the cost function.
In the longitudinal oscillatory tests of the F-16XL
aircraft model, the angle of attack varied from 0 to a
little over 70 dog. To cover this range of angle of at-
tack, we use n = 37 (.Aoc= 2 deg) for Model I and
n = 16 (Ao_ = 5 deg) for Model II. More number of
nodes were used for Model I for better definition of the
parameter variations. Further, we have to specify cer- Figure 1.Three-View sketch of the F-16XL aircraft
tain GA specific parameters such as population size, model.
probability of crossover and probability of mutation. In
this study, we use population size 30. probability of In the oscillatory tests, the mean angle of attack
crossover 0.95 and probability of mutation 0.01. If the was close to 35 deg and the oscillation amplitude was
GA reduces cost as the iterations progress, then the GA also about 35 deg so that the angle of attack varied from
was allowed to continue. Otherwise, the GA was 0 to a little more than 70 deg during one oscillation.
stopped, the values of lower and upper limits were rede- The test section speed was around 17.5 m/see that cor-
fined and the GA was started again. Since the GA has responds to a Mach number of about 0.05. The period
no defined convergence criterion to end the iteration of oscillation Tj, varied from 12 s to 1.33 s so that the
process, the GA was stopped when it became evident corresponding reduced frequency k was in the range
that further continuation would not result in any im- 0.011 to 0.101. For the purpose of parameter estima-
provement inJ. tion, the oscillatory wind tunnel data corresponding to
Tp= 2.38 s (k = 0.056) was used. The rest of the avail-
The convolution integral was evaluated using the able oscillatory test data were used for validation of
trapezoidal rule with a step size of 0.0001 s. The inte- parameter estimates.
gration of the ordinary differential equation was done
using a forth order Runge-Kutta method with a fixed The variation of the lift and pitching moment coef-
step size of 0.02 s. All the computations were done in ficients from static wind tunnel tests on the F-16XL
MATLAB. aircraft model are presented in Fig. 2. Using numerical
curve fit to these data points, the lift-curve-slope C_
Results and Discussion: and slope of the pitching-moment-coefficient C,,,awere
The above formulation for evaluating unsteady lift
and moment coefficients based on indicial response
method was applied to the analysis of longitudinal os-
cillatory data of the 0.1 scale model of the F-16XL air-
craft obtained in the Langley 12-Foot Low Speed Wind
Tunnel. A three-view sketch of the 0.1 scale model of
the F-16-XL aircraft is presented in Fig. 1. Using this
sketch, the center of the planform area ._,._was found to
be approximately at 1.0090 m. The moment reference
point is located at a distance of 0.9932 m from the nose.
The mean aerodynamic chord (mac) of the 0.1 scale
model of the F-16XL aircraft is equal to 0.753 m. Ad-
ditional information on the test model and test facility
may be found in Refs. 9 and 11.
Figure 2. Static wind tunnel data of the F-16XL
aircraft model.
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deduceadsshowninFig.3.Howevetrh,evalueosf 80
CL,,
and C,,,, deduced by two different methods of curve
70 ,_
fitting such as cubic splines and cubic polynomials dif-
60
fered by as much _+10% for CL,_and about ___15_ for
C,,,_.especially at high angles of attack. In view of this, 50
we assume 0.9 _<a_*(o_)< 1.1 in Eq. (I I) and 0.85 <_
_,deg 4O ,!
at*(cx) < 1.15 in Eq. (12). For better definition of these
error bounds, more data points taken at much smaller 30
intervals are needed. However, such data were not
20
available for this study.
10 ...... V
0.05 6
• ! i........... ! ..........._............... 0 2 10 12
0.04 Time, sec
0.03
Figure 4. Time history of angle of attack, Tp= 2.38 s.
0.02
!c i _ i i : i i
C_, Cm_0"01 o Oscillatory WT Data
- - - Model 1
per deg 0 -- Model 11
1.6
-0.01
1.4
-0.02
1.2
-O.03
-0,04 i i i * 1.0 ... ©
-10 0 10 20 30 40 50 60 70 80
_,deg CL 0.8
0.6
Figure 3. Lift-Curve-Slope and slope of pitching-
moment-coefficient of the F-16XL aircraft model. 0.4
0,2
The unknown coefficients in Eq. (15) - (17) for CL,_
and Cmq were estimated as follows: Model I: 2 4 6 10 12
cj = -0.3844, c2 = 2.3776, ca = --4.5026, dr = -0.3678, Time, sec
d2 = -0.1616, and da = 2.0361; For Model II:
c_=-0.4240, cz = 3.3127, ca = -3.3840, dj = -1.2450, Figure 5. Time history of measured and estimated
d2= -0.3806, and da= 1.5557. lift coefficient, Tp =2.38 s.
The time history of the angle of attack for o Oscillatory W-I-Data
-- - Moclel I
T, = 2.38 s (k = 0.056) is shown in Fig. 4. The esti- -- Model II
mated and measured time histories of lift coefficient are 1.6 ! ! ! _1 * ! !
shown in Fig. 5. A cross plot of Q(t) with _(t) is shown 1.4
in Fig. 6. It is observed the estimated time histories
1.2
match the measured history fairly well, with Model II
giving better fit to the oscillatory test data. A cross plot 1.0
t7 :
of the corresponding time varying lift coefficient with
CL 0.8
time varying angle of attack is shown in Fig. 6. It was
0.6
observed that the contribution of rotary derivative C_ is
very small and negligible. On the other hand, the con- 0.4
tribution of unsteady term y(t) happens to be signifi-
0.2
cant. It is quite possible that if more accurate values of
CL_ were available, Model I may have given better re-
10 20 30 40 50 60 70 80
sult. The estimated values of the indicial parameters et,deg
Model I and Model II are shown in Fig. 7(a) and 7(b).
For comparison, the lift-curve-slope is included in Fig. Figure 6. Comparison of measured and estimated
7(b). It is observed that the estimated values of a2"(_) lift coefficient, Tp = 2.38 s.
7
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areintherange0.38to0.46suggeslinthgaitheinitial
5 _ i i i
peakvalueofthenoncirculatoloryadinigsonly0.38to
0 , -
0.46ofthatbaseodnlineapristontheory.
o : : :
--5 ca ............ .'........... !........... : " : :
40 _ _ , ! ! ! , -10 ......_............_....................i.............................
600, _ _ _ _ _ _ _
4oo .............................................. .... -2Oil _ :
-25
0 I I i • _ _ l i
-30
-35 i i i i i i i
a( 09 1 .:, :, i i i_ _i .... i t 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Time, s
0.50
o,45 i t
Figure 8. Estimated deficiency function components
0.35 _ ' ' ' ' _ ' --_ in lift coefficient, Tp= 2.38 s,cc= 15 deg
0 10 20 30 40 50 60 70 80
cqdeg
5
Figure 7(a) Estimated indicial function parameters 0 _j .... r........ i
in lift coefficient, Model I.
-5
...;.............:..........:...................:.........................
,t ::::::::::::::::::::::::::::":::::::::::::::::.:.:.:.]:.:::::::: -10
. o Circulatory Component
c.I.o... ..........t.................
-15 ...... o Noncirculatory Componenl
: -- Total, Model I
a, -1 ....:..........:..........,........ : ". : ..= .i...:=.-..::- - - - Model II
-2 --i .... ;.......i.... "L-' ....................i..-.': -20 .......i............!........................:................................
...i........!.............................................
-25
-10 0 10 20 30 40 50 60 70 80
o_,deg -30
10 , _ , ,
8 ..... ,..........i.... i ...... i.......... -35 _ , _ ' _ '
0 0.05 0.10 0. 5 0.20 0. 5 0.30 0.35 L40
Time, s
Figure 9. Estimated deficiency function components
0 10 20 30 40 50 60 70 in lift coefficient, Tp= 2.38 s,cc= 30 deg.
_,deg
o
Figure 7(b). Estimated indieial function parameters
-5
in lift coefficient, Model II.
-10
Using these parameter estimates as shown in Fig. : i
-15
7, variations of the noncircular and circular components Circulatory Component
as well total loading for Model I (Eq. (l 1)), and total -20 ....... o Noncirculatory Component _
-- Total, Model/
loading for Model II (Eq.(18)) were calculated for - Model 11
-25
0 < _ < 70 deg. Some sample results of these variations
are shown in Figs. 8 to 10. As expected, the noncircu- -30
latory component decays rapidly and the circulatory -35
component decays gradually. The area under each of
-40
these curves is a measure of their contribution to the 0._)2 0.()4 0.()6 0.()8 0.'I0 0.12
unsteady lift coefficient. The calculated areas are shown Time, s
in Fig. 11. To first order in frequency, the area under
the deficiency function is related to the parameter C_ Figure 10. Estimated deficiency function
or C,,, as the case may be. Thus, the process of esti- components in lift coefficient, Tp= 2.38 s, c_=50 deg.
mating CLu and area under the deficiency function is
equivalent to estimating the combined parameter
8
American Institute of Aeronautics and Astronautics