NASA / CR-1999-209547

Modeling of Longitudinal Unsteady

Aerodynamics of a Wing-Tail Combination

Vladislav Klein

The George Washington University

Joint Institute for Advancement of Flight Sciences (JIAFS)

Langley Research Center, Hampton, Virginia

September 1999

https://ntrs.nasa.gov/search.jsp?R=19990100653 2018-05-14T13:09:40+00:00Z
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NASA / CR-1999-209547

Modeling of Longitudinal Unsteady

Aerodynamics of a Wing-Tail Combination

Vladislav Klein

The George Washington University

Joint Institute for Advancement of Flight Sciences (JIAFS)

Langley Research Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

Prepared for Langley Research Center

under Cooperative Agreement NCC1-29

September 1999
Available from:

NASA Center for AeroSpace Information (CASI)

7121 Standard Drive

Hanover, MD 21076-1320

(301) 621-0390

National Technical Information Service (NTIS)

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(703) 605-6000
Summary

Aerodynamic equations for the longitudinal motion of an aircraft with a horizontal tail were developed. In this development, emphasis was given on obtaining model structure suitable for parameter estimation from experimental data. The resulting aerodynamic models included unsteady effects in the

form of linear indicial functions. These functions represented responses in the lift on the wing and tail

alone, and interference between those two lifting surfaces. The effect of the wing on the tail was formulated for two different expressions concerning the downwash angle at the tail. The first expression used

the Cowley-Glauert approximation known as "lag-in-downwash," the second took into account growth

of the wing circulation and delay in the development of the lift on the tail. Both approaches were demonstrated in two examples using the geometry of a fighter aircraft and a large transport. It was shown

that the differences in the two downwash formulations would increase for an aircraft with long tail arm

performing low-speed, rapid maneuvers.

Symbols

A E amplitude ratio

a t, a w lift curve slope of the tail and wing

b wing span, m

C a aerodynamic force and moment coefficient

Ca(t)

C (t)

q

indicial functions of the aircraft

C L, C m lift and pitching-moment coefficient

CLw_ (t)'

CLw_(t)

indicial functions of the wing

CLt_w(t),

CLt_ (t)

indicial functions of the tail

CLt (t) gust lift indicial function of the tail

C

ma._

pitching-moment coefficient about mean aerodynamic center of the wing

c 0, c 1.... parameters in indicial functions

_-' mean aerodynamic chord, m
d01 , d I.... spline parameters in indicial functions

F_(t),

Fa (t)

FLtaw(t),

FLtat(t)

Ge(ik)

Im( )

L

_t

q

Re( )

S

V

g t

1

deficiency functions

deficiency functions of the tail

deficiency functions of the wing

frequency response function

distance between a.c. of the wing and aircraft c.g. as a fraction of mean aerodynamic

chord

imaginary part of complex number

imaginary unit, i = ,,_

reduced frequency, k = co_/V

a) distance from bound vortex to reference point, m

b) lift, N

distance from wing trailing edge to tail leading edge, m

tail arm, m

rate of pitch, sec -1

real part of complex number

,-)

wing area, m"

time, sec

airspeed, m/sec

tail volume, V t = St-fitS?

induced vertical velocity, m/sec
F

A

eCL (t),ec_( t )

A

CO

Superscript:

Subscript:

t

W

distance from reference point to starting vortex, m

angle of attack

circulation, m2/sec

increment

time delay between harmonic responses, sec

downwash angle, rad

downwash indicial functions

aspect ratio, A = b'/S

exponents in indicial functions, sec -I

air density, kg/m 3

time delay, sec

phase angle, rad

angular frequency, sec-I

time derivative

Fourier transform

nondimensional quantity

"lag-in-downwash" formulation

tail

wing

Derivatives of aerodynamic coefficients Ca, where index a = L or m

3
OC OC.

c%coo)-c aq = qC =

Introduction

Recently there has been an increased interest in identification of aircraft aerodynamic models that

include unsteady aerodynamic terms. The majority of identified models were related to the longitudinal

motion of tailless aircraft, see reference I to 3, where the aerodynamic models had relatively simple

form. In formulating aerodynamic model equations for a wing-tail combination, however, more complicated model forms can be expected. This problem was first addressed in the early twenties by Cowley

and Glauert (ref. 4). They realized that there was a time lag before the wing disturbance reaches the tail.

They assumed that the downwash associated with a change in lift is equal to the corresponding steady

value but the effect at the tail is delayed by the time for the airplane to travel a distance equal to the tail

arm. The change of the lift on the tail due to downwash was included in the damping-in-pitch derivative. Now this change is interpreted as a tail contribution to the acceleration derivatives, i.e., the derivative of the lift and pitching moment with respect to rate of change in the angle of attack.

The investigation of the downwash angle was extended by R. T. Jones and Fehlner in reference 5 by

considering both the growth of wing circulation and the delay in the development of lift by the tail.

They presented an expression for the downwash indicial function associated with a change of the lift on

the wing. Considerable effort to the rigorous explanation of the downwash both qualitatively and analytically was given by Tobak in reference 6. He developed indicial functions for the lift and pitching

moment of a wing-tail combination and presented numerical results for several representative cases in

supersonic flight regimes. The theoretical results were then applied to the longitudinal stability analysis

of an aircraft.

In reference 7 a theoretical analysis of the downwash due to changes in the lift of the wing was presented. In this analysis the wing was approximated either by a single horseshoe vortex or by a set of

these vortices. The result showed that for specific changes in the lift the downwash would depend on the

tail arm, wing aspect ratio and its span. Similar to reference 7, a simple vortex system was used in reference 8 to model unsteady aerodynamic effects into the longitudinal equations of motion. It was found

that the solution of resulting integro-differential equations led to lengthy computations. The calculations

also showed that the unsteady aerodynamics could be adequately accounted for when the unsteady

effects were included only in the downwash development. This finding permitted reformulating the

equations so that they appeared suitable for parameter estimation using flight test data.

This report is an extension of previous research in references 5, 7, and 8. Its purpose is to develop

aerodynamic equations for the longitudinal motion of an aircraft with horizontal tail. These equations

include unsteady effects modeled by linear indicial functions. The emphasis is given to obtaining model

structure suitable for model identification using experimental data. It means that the postulated model

should explain the aerodynamics of wing-tail combination with sufficient accuracy and, at the same

time, should be simple enough to provide good conditions for parameter identifiability. The report starts

with the formulation of equations for the lift and pitching-moment coefficient and introduction of indicial functions in these equations. Then the development of an indicial function relating the change in the

lift of a wing due to downwash is presented. The downwash angle during the harmonic motion of an aircraft is compared with that based on the traditional "lag-in-downwash" factor contained in the acceleration stability derivatives. The variation of the downwash angle with the geometry and frequency of the

harmonic motion is also briefly discussed. Next, the final form of aerodynamic model equations is pre-
sented.Numericalexamplesusingthegeometryof afighterandtransportaircraftdemonstratepossible

formsof indicialfunctionsusingboththe"lag-in-downwash"formulationandthatwithunsteadyterms.

Thereportiscompletedbyconcludingremarks.

Indicial Functions

In the following development the lift and pitching moment will be considered as functions of the

angle of attack and pitching velocity, i.e.,

Ca = Ca(_,q), a=Lorm

Then the aerodynamic model equations can be formulated as

t t

?.

c,,(t) = c(o) + c%(t - + (t -

o o

(1)

Where Ca(O) is the value of the coefficient at initial steady-state conditions, and C(t) and Car(t)

are the indicial functions defining the responses in Ca to unit step in _ and q respectively. Replacing

the second indicial function in ( I ) by its steady value Caq(_), equation ( 1) is simplified as

t

Ca(t ) = Ca(0 ) + -_-_C%(oo)q(t) + fCa (t- "c)d_("c)d_ (2)

0

When the deficiency functions

F%(t) = Ca (o_)- Ca (t) (3)

are substituted into (2) the equations for the aerodynamic coefficients take the form

I

Ca(t) = Ca(O) + Ca,(°°)ot(t) + __" Ca,,(°°)q(t) ; Fa (t _.c)6t(.c)d. c -

0

(4)

where Ca,(',o) and Caq(oo) are the rates of change with ot and q of the coefficients C L and Cm .

The indicial functions C a (t) include combined responses of the wing and tail, and interference

effects between those two lifti_ag surfaces. Because of linear aerodynamics the resulting indicial functions are given as a sum of their components. For the indicial function CLJt) these components will be

as follows:

( I) the response in the lift of the wing to a unit step in angle of attack of the wing, while the angle of

attack of the tail remains at zero;

(2) the response in the lift of the tail to a unit step in angle of attack of the tail, while the angle of

attack of the wing remains at zero;
(3) theresponseinthelift of thetail toaunitstepintheangleof attackof thewing,whiletheangle

of attackof thetailremainsatzero;

(4) theresponsesinthelift of thewingtoaunitstepintheangleof attackofthetail,whiletheangle

of attackof thewingremainsatzero.

The first two componentsarerepresentedby theresponseof anisolatedwingandisolatedtail

respectively.As pointed out in reference 6, these representations are true for supersonic speeds. For

subsonic speeds, however, these are good approximations. The third component expresses the lift on the

tail due to a change in the downwash induced by the lift of the wing. It is usually a significant contribution to the resulting pitching moment of an aircraft. Finally, the last component is zero for supersonic

speeds. For subsonic speeds it can be neglected. A summary of all four components, their conditions

and interpretations is given in table I.

The expressions for the indicial function CLw_(t) of a wing alone were given by several authors.

In reference 5, the indicial function was formed as a sum of exponential

CLwa.(t) = c O + Zcje (5)

J

where the values of parameters c,, c. and _,. depend on the wing geometry. For a wing with small

v j j

aspect ration equation (5) was further simplified as

CLw,_, (t) = c o + c 1e (6)

Equation (6) was used in references 1 and 3 for aerodynamic model identification of tailless aircraft

from wind tunnel, small amplitude oscillatory data. In reference 2, the parameters in (6) were considered as functions of the angle of attack for the identification of nonlinear models from wind tunnel

dynamic data with large amplitudes. In both cases, identified models were found to be adequate representations of experimental data.

A simple form of the indicial function CLt_,,(t) defining the downwash effect on this tail can be

obtained by considering that a sudden change m the angle of attack of the wing results in a sudden

change in the downwash. Downwash from the wing is delayed from reaching the tail by the time

£t

At = --

V

where "gt is the distance from the a.c. of the wing to that of the tail and V is the airspeed that remains

constant and equal to the speed of undisturbed flow. By neglecting the unsteady aerodynamic effect on

the tail, the indicial function CLt,_w(t ) can be expressed as

CLtaw(t) = CLla,(,,o)(t- At) (7)

Its time history is sketched in figure l a. This assumption about the downwash has been used by Cowley

and Glauert in formulating contributions of the tail to the acceleration stability derivatives CLa and

C ttl ot "

A more detailed qualitative assessment of CLt_,(t ) is given in reference 6. At the time of sudden

change in ocw and under condition oct = 0, no disturbances are present at the tail. At certain time

tI > 0, the tail begins to penetrate the wake created by the vorticity shed by the wing. The tail begins to

6
developpositivelift onthatportionof its surface that has penetrated the wake. As time increases the tail

further penetrates the field of shed vorticity and is subjected to a combined effect of upwash and downwash. Further penetration into the wake results in a change of lift from positive to negative. At certain

time t 2 > t I the tail is fully immersed in the steady downwash field. Then the lift on the tail remains

constant because it is no longer influenced by the disturbances created by the wing at the beginning of

the motion.

From the preceding discussion one can anticipate the shape of CLt_(t) as sketched in figure lb

and its analytical form as a combination of three effects:

a) change in the lift of the wing followed by a change in angle of attack of the wing;

b) change in the downwash at the tail following a change in lift on the wing;

c) change in the lift on the tail due to a change in downwash at the tail.

Therefore, the variation in the lift on the tail following a change in angle of attack of the wing, while

maintaining s t = O, follows as a solution of three equations

t

CLw(t) = Ct.w._,(t)c_(O) + ICLw_,,,(t - "c)a('t)d't (8)

0

t

c(t) = CCL(t)CLw(O) + ICcL(t -'t)CLw('t)d't

0

(9)

l

CLt(I ) = - CLt_,(t)c(O ) - ICLt (I - Z)_(x)d'c

0

I0)

The minus signs in equation (10) reflect the definition of a positive downwash angle that results m a

decrease of the angle of attack at the tail. The corresponding indicial function CLtt_,(t) can be obtained

from equations (8) to (10) as

!

ca(t ) = CCL(t)CLwc_w(O) + IECL(t - _)OLw,,('c)d'c (I 1)

0

!

C Lt,_, (t) = - C Lt ,( t)£o_(O) - I C Lt._(t - "t)p.a('t )d'c

0

where the last equation can be also written as

7

(12)
t

CLt_,,(t) = - ect(t)CLt (O) - _eot(t - "C)CLt ('c)dT

0

(13)

In the above equation the indicial function CLt_,.(t ) has already been discussed. The function CLt(t )

represents the lift on the tail during passage of the tail through a step change in downwash that is equivalent to the passage through a sharp-edge gust. The two-dimensional theory for this case was developed

by Kiisner in reference 9 and extended to the three-dimensional case in reference 10. It was found that

the indicial function CLt(t) can again be approximated by a sum of exponentials similar to equation

(5) for CLtc_, (t).

Analytical expressions for the indicial function ECL(t ) in equations (9) and ( 11) were developed by

several authors, see e.g,, references 5, 7, and 8 using various degree of approximation to the vortex

sheet behind the wing. A rather simple approach to this problem is demonstrated in the next section.

Once the indicial function explaining the variation in lift of an aircraft following changes in the angle of

attack are known, they can be converted to those in the pitching-moment equation.

Downwash Angle

In the following development the vortex sheet of a finite wing in an incompressible steady flow is

replaced by a horseshoe vortex consisting of the bound vortex and two trailing vortices. After a sudden

change in the circulation around wing, a stationary vortex separates from the trailing edge of the wing

and moves in the direction of the velocity vector with the velocity equal to that of free stream, see

figure 2. The vortex system shown in figure 3 indicates the position of a reference point A and distance

between trailing vortices. For the present study this distance is equal to the wing span. As pointed out in

reference 11, however, the distance between trailing vortices is less than the wing span. It can be determined by equating the lift of the replacement system to the lift of the wing.

The downwash angle at the point A is e = wi/V where wi is the sum of induced velocities by all

four vortices. These velocities are calculated in the plane of the vortex system, midway between the two

trailing vortices, from the Biot-Savart rule. The expression for the downwash angle takes the form

b/2

E(x) AF x + b/--_ + T

L4t, T) + x ] + L 2

where AF is the circulation strength of each four vortices and L = 3/4b +,g. It is convenient to

express the downwash angle in nondimensional form as a function of time. This can be achieved by

considering the relation

x = Vt-_ p (15)

where _' is the distance between the wing trailing edge (the origin of the wing circulation) and the leading edge of the tail. By introducing nondimensional time

8
andnondimensional lengthsintermsof the half span, i.e.,

x b/2 '

..g L

- -- and L' = --

b/2 b/2

equation (15) takes the nondimensional form

(16)

x' = .g'(t'- I) (17)

Substituting (17) into (14) and replacing L by L' yields

It is further assumed that the Zhukovski's formula

I "_

9VAFb = _gV'SAC L

can be also applied to the unsteady flow around the wing. This leads to an increment in the circulation

6r - sv ac

2bA L (19)

where A = b2/S is the wing aspect ratio. Combining (18) and (19) gives the expression for the downwash indiciai function

and it's steady value

 CLt/) : + l ) +--

(20)

(21)

The indicial function ECL(t') is sketched in figure 4, It shows infinite values for t' = 1 which is the

time when the leading edge of the tail reaches the position of the starting vortex. After that, the downwash angle decreases and, with increasing time, approaches a steady value. Although the indicial function can reach an infinite value, the convolution with the lift results in finite downwash at all points. If

the wing wake passes either above or below the tail surface, the indicial function ECL(t" ) will not have

infinite values. Using the Cowley-Glauen explanation of the downwash delay and considering L = "gr

the expression for the downwash indicial function is simplified as

-- <22,

9
or

(23)

where l(t -L) and l(t'-_)

figure 4 as a dashed line.

are the unit step functions. The time history of ECL(t ) is included in

A difference in calculated downwash using indicial function given by equation (20) or (23), and the

effect of geometry and flight condition will be demonstrated on the harmonic motion of an aircraft.

Equation (9) has its frequency domain equivalent

_(ik) = ik_cL(ik)CLw(ik) (24)

which leads to the frequency response function

_(ik)

Ge(ik) - -_Lw(ik )

where k = o3.g/V is the reduced frequency and

- ik_cL(ik) (25)

_CL(ik) = ECL(t , )e -ikt' dt ,

(26)

For the numerical evaluation of the Fourier integral in (26) it is advantageous to express the indicial

function ECL(t') in the form

ECL(t" ) = _CL (o_) - FE(t" )

where the deficiency function follows from (20) and (21) as

1

Ve(t') = _ I _/I +.ff'2(t'- !) 2 '1 (27)

Using equations (25) and (27) the frequency response functions relating the downwash and the lift takes

the form

Ge(ik) = _CL (_) - ik_Fe(t')e-ikt'dt '

0

(28)

The computed frequency response curve 2xAGe(ik) for 4' = 1.0 and L' = 1.2 is plotted in figure 5

as a vector diagram. In the same figure the frequency response curve based on the Cowley-Glauert's

"lag-in-downwash" formulation

G*(ik) = ECL(Oo)e -ikL/£ (29)

10
is alsogiven.Forthepresentationof differencesin bothdownwashformulationsandtheir variation

withthefrequencyanddistance,g', the transfer functions Ge(ik) and G_(ik) defined by the previous

two equations are expressed in terms of amplitudes and phase angles as

icp

Ge(ik) =Aee E (30)

a* (ik) = (31)

Then the amplitude ratio AJA* for three different values of _' = 1.0, 0.5 and 0.25 are plotted in

figure 6. Figure 7 shows the time delay between harmonic responses obtained from equations (28) and

(29) as

q_E L

8' - (32) k £

Considering the results in figures 5 to 7, the differences in both downwash formulations depend mainly

on two parameters, the reduced frequency, k, and nondimensional distance _'.The effect of nondimensional tail arm L' = _ is rather small as follows from equations (20) and (21). The differences in

amplitude ratios increase with increased reduced frequency. Increased reduced frequency corresponds

with rapid maneuvers at low airspeed for an aircraft with long tail arm. On the other hand, these differences at given reduced frequency decrease with increased parameter _'. As to the differences in the time

delays, an increase in reduced frequency makes these differences, in general, smaller. At a given

reduced frequency the differences in time delays will decrease with the parameter _' which is especially

apparent at low values of k, see figure 7. More detailed assessment of both downwash formulations can

be made only for specific aircraft geometry and flight conditions.

Lift and Pitching Moment

Returning to equation (4) the lift and pitching-moment coefficient are expressed as

t

CL(t ) = CL(O ) + CL (_)_(t ) + 9_CLqq(t) -_FLjt - x)6t(x)d'c (33)

0

!

Cm(t) = Cm(O) + Cm (oO)c_(t) + _--_C,,,uq(t) - _F,,, (t - "t)_(z)d'c

0

(34)

Contributions of the wing and tail to these coefficients can be found from the relations

S I

CL(t ) = CLw(t) - -_CLt(t )

Cm(t) = Cm,,. ' + hCL(t)- VtCLt(t )

(35)

(36)

where Cm,, is the pitching moment about the aerodynamic center of the wing, a.c., h is the distance

between the a.c. and the aircraft center of gravity expressed as a fraction of ?, V t = SiftS? is the tail

volume and CLl is based on S t .

11
Combing equations (33) through (36) yields

?.

CL(t ) = CL(O )+CL (oo)at(t)+_CLq(oo)q(t)

t

S t

-f[FLw_,,(t- x)+ -_FLt,_,(t- _)

o

Cm(t) =

t

+ St .t )16t ( _ )d. c "_FLt,,(t -

_C

Cm(O ) + C,,,,(oo)at(t) + _ ,,,,(oo)q(t)

(37)

(38)

-_i hFLwa_,(t - "c)-VtFLt_ (t - "c)-VtFLta (t - "C)16_('c)dx

o

where the deficiency functions are defined as

FLw_,(t) = CLw,_,(oo) - CLw_,(t) (39)

FLt,_ (t) = CLr_r(oo)- CLt_r(t) (40)

FLtc_ (t) = CLtaw (oo) - CLtc_,,(t) (41)

As mentioned earlier, the steady parts of indicial functions in equations (37) and (38) are aircraft stability derivatives CL, Cm. CLu and Cmu. The first two are usually formed as

CL_ = a w I +-- I-_-_ (42)

a W

Cm, _ = a w h- a--w Vt 1--ff_

(43)

where the contributions of the wing and tail are easily recognized. In these expressions

a w = CLw_,(oo) and a t = CLt_(oo) are the lift curve slopes of the wing and tail respectively.

The derivatives CLq and Cmq represent aerodynamic effects that accompany rotation of the aircraft

about its lateral axis while at remains constant. Both the wing and tail are affected by the rotation but the

wing contribution is usually negligible. A discussion of the damping term Ctrtq due to wing in supersonic flight conditions is contained in reference 6. The contribution of the tail includes changes in the

lift on the tail due to increased angle of attack by (q.gt)/V and changes due to effective angle of attack

at the tail caused by the pitching velocity of the wing. The latter change in the lift is discussed in references 12 and 13 for supersonic flight regimes. Thus considering a contribution of the tail only, the

damping derivatives are approximated as

CLq -- 2a t V t (44)

12
Cm =-2aft. V t (45)

q

Examples

In the following examples, the indicial functions CL (t) and Cm (t) were computed using two sets

of data representing a fighter and transport aircraft. The _eometrical _haracteristics of these aircraft are

summarized in table 2. The approximate expressions for the indicial functions of the wing and tail,

CLw (t) and CLt (t) , and the gust lift function CLt (t) were taken from reference 14. The func-

(X . t t,, .

tions°_or given aspec_ ratnos and argument t = Vt/_g have the form

for fighter aircraft:

C ' 3.77(I 0.283e -°626t') Lwf_w(t ) =

CLtc_r(t') = 4.65( I - 0.361 e -0"442t')

CLt (t') = 4.65( I - 0.448e -0"336t' - 0.272e -°841 _'

,t_

for transport aircraft:

-'_ 48t',

- 0.193e " )

s

CLw_,(t') = 4.65(1 - 0.361e-12t )

CLt_r(t') = 3.77( 1 - 0.283e -22°t')

CLt(t , -_'_ ' 8.5t ) ) = 3.77(1 -0.679e '_'3t _ 0.227e-I '

The indicial functions eCL(t' ) and eCL(t" ) were computed from equations (20) and (23), indicial function CLtc. (t') from equations (l l) and (13). The resulting indicial functions CL(t" ) and C,n,(t" )

were obtained from equations (35) and (36) as

assuming Cm,,.. = 0.

S l

CLJt') = CLw_w(t') + _(CLtc_,(t') + CLt_w(t')) (46)

CmJt') = hCLa(t')-VT(CLt_ (t') + CLt_.(t')) (47)

The three indicial functions of the fighter aircraft appearing on the right side of equations (46) and (47)

are plotted in figure 8, the functions CL(t' ) and Cm, (t') in figure 9. The dashed lines in these figures

indicate the "lag-in-downwash" approximation of the indicial function eCL(t'). Figure 10 and I 1

present the same functions as above for the transport aircraft. As expected, the form of the indicial functions CL(t' ) for both aircraft was only slightly affected by different expressions for ECL(t'). This

means that the model structure for CLa(t" ) can be postulated as that for the wing alone, i.e.,

13
CL (t) = co + _jcje X)t

The structure for an adequate model for C m (t'), on the other hand, could be strongly affected by

p . . O_ ....

e_CL(t ), evaluated either from equation (20) or (23). As mentioned previously, the decision between

using one of the forms for ECL(t') will depend on the values of two parameters, k and ,g'. Assuming a

harmonic motion with _ = 4 rad/sec and V = 100 m/sec, and given geometry, then for the fighter aircraft

k = 0.08 and .g" = 0.44. Using figures 6 and 7 it can be found that the amplitude ratio, Ae = 1.0 and

time delay 8--- 0.017 sec. In this case, therefore, the Cowley-Glauert approximation for ECL(t" ) would

be warranted. For the transport aircraft k = 0.96 and ,g' = 0.80 which resulted in A t ---2.0 and

8 = 0.12 sec. In this case, equation (20) for ECL(t' ) should be used. Then the model for Cm(t') can be

postulated as

0

Cmjt ) = d O+d l(t-t 1)+ (49)

or

C,,, (t) = d o + dlt+ d2(t- t 1)+ + d3(t- t2) +

(50)

The plus functions in (49) and (50) are defined as

o 1 fortj >t

(t-tl)+ = Ofort l<t

t-t I fort I >t

(t-tl) + =

0 fort I <t

and similary for (t - t2) + .

Equation (49) reflects sudden change in the downwash angle, whereas equation (50) takes into account

gradual changes in the downwash. In equation (49) the value of d o would be different from that based

on Cowley-Glauert's approximation of ECL(t). The time t 1 in both expressions might be close to

t = .gt/V.

Concluding Remarks

Aerodynamic model equations for the longitudinal motion of an aircraft with horizontal tail were developed. The equations include unsteady effects modeled by linear indicial functions. These functions represent responses in the lift and tail alone, and interference between those two lifting surfaces. The

indicial functions for the lift on the wing and tail were modeled by a series of exponential functions. The

effect of the wing on the tail was formulated for two different expressions for the downwash angle at the

tail. The first expression used the Cowley-Glauert approximation known as "lag-in-downwash." The

second took into account growth of the wing circulation and delay in the development of the lift by the

tail. From analytical expressions obtained and numerical examples, the following conclusions can be

drawn:

a) for harmonic motion the differences in calculated downwash using simple "lag-in-downwash"

concept and that with unsteady aerodynamics depend upon the frequency and airspeed of the

motion, on the distance between the wing trailing edge and tail leading edge, and on the wing

14
span. These differences will increase for an aircraft with long tail arm performing rapid maneuvers at low speed. More detailed assessment of both concepts, however, can be made for specified aircraft geometry and flight conditions;

b) a model for the lift is only slightly affected by different approaches in computing the downwash

angle. For the preliminary studies and model identification, the model structure can be the same

as that for the wing alone;

c) in pitching-moment formulation the "lag-in-downwash" approximation is warranted for aircraft

with short tail arms. For aircraft with long tail arms, however, the expression with unsteady

terms should be considered. Then the resulting indicial function for the pitching moment can be

approximated by zero-order or first-order splines.

References

.

Klein, Vladislav: Murphy. Patrick C.; Curry, Timothy J.: and Brandon, Jay M.: Analysis of Wind Tunnel Long#

tudinal Static and Oscillatopw Data of the F-16XL AiJz'raft. NASA/TM-97-20676. 1997.

Klein, Vladislav: and Murphy, Patrick C.: Estimation of Aircraft Nonlinear Unsteady Parameters From Wind

Tunnel Data. NASA/TM- 1998-208969, 1998.

3.

,

5,

Smith, Mark S.: Analysis of Wind Tunnel Oscillatory Data of the X-3 IA Aircraft. M.S. Thesis, Thc Georgc

Washington University, Washington, D.C., 1998.

Cowley, W. L.; and Glauert H.: The Effect of the Lag of the Downwash on the Longitudinal Stability of an Airplatte and on the Rotary Derivative Mq. R. & M. 718, Feb. 1921.

Jones, Robert T.; and Fehlner, Leo F.: Transient Effects of the Wing Wake on the Horizontal Tail. NACA TN77 I, 1940.

6.

.

8,

Tobak, Murray: On the Use _!f Indicial Function Concept in the Analysis _f Unsteady Motions _f Wings attd

Wing-Tail Combinations. NACA Rep. I 188, 1954.

Danek, M.: Downwash Angle at the Horizontal Tail During Rapid Oscillations of an Aircraft. Internal Report,

Military-Technical Academy (VTA), Brno, Czech Republic, 1956.

Queijo, M. J.; Wells, William R. and Keskar, Dinesh A.: Inclusion of Unsteady Aerodynamics in Longitudinal

Parameter Estimation From Flight Data. NASA TP- 1536, 1979.

9. Kiissner, H. G.: Zusammenfassender Bericht fiber den instationfiren Auftrieb von Fltigeln. Luftfahtforshung,

Bd. 13, Nr. 12, Dec. 1936, pp. 410-424.

10. Jones, Robert T.: The Unsteady Lift of a Wing of Finite Aspect Ratio. NACA TR No. 681, 1940.

I I. Krasnov, N. E: Aerodynamics. NASA q'T F-765, 1978, pp. 234-239.

12. Ribner, Herbert S.: Time-Dependent Downwash at the Tail and Pitching Moment Due to Normal Acceleration

at Supetwonic Speeds. NACA TN-2042, 1950.

13. Martin, John C.: Diederich. Margaret S.: and Bobbitt, Percy J.: A Theoretical Investigation of the Aerodynamics _f Wing-Tail Combinations Petfiwming Time-Dependent Motions at Supersonic Speeds. NACA TN-3072,

1954.

14. Jones, Robert T.: Tire Unsteady Lift of a Wing of Finite Aspect Ratio. NACA Rep. 681, 1939.

15
TableI. Summaryofcomponents ofindicialfunctionsexpressingchangesinlift due

tounitstepinangleofattack.

Indicial

Component Function Conditions Interpretation

I CLwc_w(t) °tw = l(t), ott = 0 lift response of isolated wing

2 CLtut(t) Otw = 0, _t = l(t) lift response of isoloated tail

3 CLtc'w(t) Otw = l(t)'Ott = 0 liftdueto downwash

4 CLw_'(t) Otw = 0, 0tt = l(t) lift due to fl t

(neglected)

Table 2. Geometrical characteristics of two

aircraft.

Characteristic Fighter Transport

2 27.9 511.0 S,m

b. m 9.14 60.0

_', m 3.45 8.3

A 3.0 " 7.0

2

S t, m 4.55 142.0

b r m 5.49 22.0

('t' m 1.45 6.4

A t 6.6 3.4

L, m 4.59 30.0

2, m 2.00 24.0

V t 0.22 1.0

h 0.05 0.05

16
CLtmv

It/V

I

t, see

(a) Downwash delay.

CLlop a

I t. sec

CLt{x_ ( `x' }

fb) Gradual vortex sheet penetration.

Figure I. Qualitative time histories of tail lift caused by unit step in wing angle of attack.

--'.11,.-

nd vortex Starting vortex_ F

Figure 2. Location of bound and starting vortices.

17
Bound

vor02x -._ -- --

/

Z L

//_Starting vortex

_,v _

x _ Trailing vortex

Figure 3. Vortex system used tor computing downwash.

I_CL(t')

0 x 1

\

I

|

_',-"_..... _CL,=_

t' = tV/I

Approximation

to e_e_t')

Figure 4. Sketched time histories of downwash indicial function and its approximation.

18
2rcA Re (G c) 0

-2

-4

-6

I

I I I I I I

.......... 7 .......... :............ '_ .......... i ........... : ........... .I ........... '. ..........

k = 3.0,,i i i ': i !

.......... ' .... r

_ {fn_,ieaa_; t:Sr.mi_,ii,,;i ........... i...................... ;........... J

L ..... 5+

:Cowley-G auerl's ':_ /

::a.,ro×/mafion--_:Z /

! ........ i i ! _...i _+,,._. #.oi 5 i....................... :........... "]'_ " ..........

l I I I I t

2 - I 0 -8 -6 -4 -2 0 2 4

2_:A im (G c)

Figure 5. Frequency response curve relating downwash at the tail and lift on the wing.._ I = ! .0

14 / i!

121 ........ !:......... +......... :......... !......... :......... : ........ :........ :"--"......... l:i __;/ :........

i i : i i i r _7+: / / ,

A__z_r

.........::.........!.........!.........i.........i........ i...... i.........!_._s_ .......

4 .........i......... :_.........::........i.........i.........i....... i.........i.........::........__

i i :, i :, :: :: i l.oi |

1

2 ......... i....... i ......... : .............. i....... : ...... ' : : ...

I I I I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0

k

Figure 6. Amplitude ratio of harmonic downwash from unsteady and Cowley-Glauert's fl)rmulation.

19
1.4

1.2

1.0

_' .8

.6

.4

.2

0

/'=0.25 i

1 I I i I

.2 ,4 .6 ,8 1.0 1.2 1.4 1.6 1.8 2.0

k

Figure 7. Difference in time delays of harmonic downwash from unsteady and Cowley-Glauert's formulation.

2O
............ [ ............. _............. , ............ , ............ ! ............. [ ............. :............. ;. ............ ;. ............

3 ------_-.-----+ +i ............ :............ :............. i ............ i ............

.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

t' = tV/I

5.0

Cl+tett

5

............ ,.............. , ............. i ............ i ............. i ............. i............. '............. +........... •.............

I

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

t' = tV/l

CLl0tw

3

2

1

0

-1

I i I

Cowl:ey-Glauert':s approx.

............ i............ ............ i............ i............

............ ?.......... -......._............. -.....i_ ......i............. i.......... i.......... i.........

: : : : __ _ ._...... :....... _ _ _ _._.5... ............ . ....

__ i i i i i i i i i i

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

t' = tV/I

Figure 8. lndicial functions of wing and tail. Fighter aircraft.

21
CLo¢

I

o

................................................... : ......................... : ........................... 7 ............

.5 1.0 1.5 2.0 2.5 3.0 3.5

t' = tV/I

...... !............. i.............

i

4.0 4.5 5.0

Cnll_

i ! i i i i i i

01 I I I I I I I I

-.2 L ........... ::............. {............. ::............. :....... ?. a £-'-" ::-'-'_:: ............. ::............. i ............ i ............

I i i i , : : --_ ...... :-..... - ......

-.4 ........... ':-............. ::............. i............. ::....... : ..... !-- :,............ ::............. '............. :.......... 1'7-

-.6 _a....... Z.2:'.r.:..... +::= _.=. f_y-Glauert's .... !............. !....... ,.... i............ ' ............. i............. approx. ::! ............ i!............

............ !\... ......... ......................... i............ i...........

-_.o ........................ ii.................. \. _......... i :: i ........... i............. i............. i............ _............

t_ i \i :, i i i i i

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

t' = tVll

Figure 9. Indicial functions of wing-tail combination. Fighter aircraft.

22
6

5

4

3

2

1

0

T

........... i.............. , .............

[ I I

.5 1.0 1.5

I [ l I

2.0 2.5 3.0 3.5 4.0 4.5 5.0

t' = tV/l

CLlott

5

4 ............ ; ............. ,............................................................................................................

3 _ ............ _...........................................................................................

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4,5 5.0

t' = tV/l

C[.tot w

-1

-2

0

 COWeyOaUe approx

I I I I I

.5 1.0 1.5 2,0 2.5 3.1) 3.5 4.1) 4.5

t' = tVII

5.0

Figure 10. Indicial functions of wing and tail. Transport aircraft.

23
CLo_

_) ............ _ ............. i ........................... '............ J ............. ' ............. '.............. • ............ '.............

1 I I I I

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

t' = tVll

Cm_

0 i r i 1 [ i [

iii ii ..... iill 1t

3 "/'_-'---i----- "_ Cowle Glauert s a rox

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

t' = tVll

Figure 1 I. Indicial functions of wing-tail combination. Transport aircraft.

24

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September 1999 Contractor Report

4. TITLE AND SUBTITLE

Modeling of Longitudinal Unsteady Aerodynamics of a Wing-Tail

Combination

6. AUTHOR(S)

Vladislav Klein

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

The George Washington University

Joint Institute for Advancement of Flight Sciences (JIAFS)

Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23681-2199

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NCC 1-29

522-33- I 1-05

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AGENCY REPORT NUMBER

NASA/CR- 1999-209547

11. SUPPLEMENTARYNOTES

Langley Technical Monitor: James G. Batterson

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13. ABSTRACT (Maximum 200 words)

Aerodynamic equations for the longitudinal motion of an aircraft with a horizontal tail were developed. In this

development emphasis was given on obtaining model structure suitable for model identification from experimental

data. The resulting aerodynamic models included unsteady effects in the form of linear indicial functions. These

functions represented responses in the lift on the wing and tail alone, and interference between those two lifting

surfaces. The effect of the wing on the tail was formulated for two different expressions concerning the downwash

angle at the tail. The first expression used the Cowley-Glauert approximation known.as "lag-in-downwash," the

second took into account growth of the wing circulation and delay in the development of the lift on the tail. Both

approaches were demonstrated in two examples using the geometry of a fighter aircraft and a large transport. It was

shown that the differences in the two downwash formulations would increase for an aircraft with long tail arm performing low-speed, rapid maneuvers.

14. SUBJECT TERMS

aircraft longitudinal motion, unsteady aerodynamics, indicial functions

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