A linearized theory for the unsteady motions of a wing in curved flight

(1)

DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER BETHESDA, MARYLAND 20034

A LINEARIZED THEORY FOR THE UNSTEADY MOTIONS

OF A WING IN CURVED FLIGHT

by

E. C. James

APPROVED FOR PUBLIC REEASE: DISTRIBUTION UNLIMÍTED

(2)

TABLE OF CONTENTS ABSTRACT

...

ADMINISTRATIVE INFORMATION

... '.

."-. INTRODUCTION KINEMATICS DYNAMICS,. u Page 2 4 BOUNDARY VALUE PROBLEM FOR ThE COMPLEX ACCELERATION

POTENTIAL ...

. 7

LEADING EDGE SINGULAR FORCE' . '.. . . . .. .. .

...

8

FORCE AND MOMENT .

. ...

9

'ENERGY AND POWER ' iO

CASE OF CONSTANT CURyATURE K

...

11

CONSTANT FORWARD SPEED V ABOUT A CIRCULAR ARC

TRAJECTORY 14

IME HARMONIC MOTION WITH CONSTANT V AND CONSTANT K 15'

CONCLUDING REMARKS 16

ACKNOWLEDGEMENT . .

ii

APPENDIX - DERIVATION OF ThE ENERGY BALANCE EQUATION 19

LIST OF FIGURES

Figure 1 - Description of Coordinate Systems. apd the Flight Path . . 3 Figure 2 - Transverse Disp1acrnent Relative to the Flight Path of a

Typical Wing Section 3

(3)

ABSTRACT

A linearized theory is developed to treat small-amplitude unsteady motions of a wing in curved flight at variable loôal forward speeds in an inviscid incompressible fluid. The wing

geometry, motions, and flight path are specified and the problem is to obtain the time-dependent

force, moment, power required to sustaiñ the motión, pressure and velocity fields, and energy

loss due to the shedding of vorticity. The theory is expected to provide useful estimates

provided the wing does not cross its own wake.

The effect of path curvature is of particular interest ¡n this investigation.. To cóñtrast this effect, the results can be readily compared with those for a wiñg in straight-line flight.

ADMINISTRATIVE INFORMATION

The work reported herein was sponsored by the in-house independent exploratory development program of'the Naval Ship Research and Development Center. Funding was provided under Program Element 61 152N, Task Area ZRO23OIO1, Work Unit l-1564-00!.

This material was presented at the AIAA 11th Aerospace Sciences Meeting held ¡n Washington, D.C.,

9-11 January 1973.

INTRObUCTION

The purpose of the report is to present a consistent first-order unsteady hydromechanical theory for an infinite aspect ratio wing in curved flight. A two-dituensional analysis in a plane transverse to the span gives good quantitative estimates of the flow för large aspect ratio wings or wings with spanwise end plates.

The investigation proceeds from the viewpoint of unsteady airfoil theory and adopts many of the ideas

exemplifled in the Wu treatise on the hydromechanics of swimming propulsion1 and the James thesis on cycibidal propulsion.2

For purposes of the present investigation, the wing speed is assumed to be small enough so that the fluid can be treated as incompressible. However, the characteristic Reynolds number, based ón wing sp'ced

and chord length, is assumed to be large. The presence of a boundary layer along the wing surface is cn-fined to a narrow region and further manifests itself in à thin wake region downstream of the trailing edge.

The boundary layer proper is neglected, and a free vortex sheet is taken to represent the effects of viscosity

in the wake. Thus the problem is one of a potential flow.

A fundamental assumption is that the wing presents itself to the quiescent flúid only at small angles

of attack as its pursues some prescribed path. Hence, the path curvature as well as the transverse motions

'Wu, T.Y., "Hydromecharücs of Swimming Propulsion, Part 1: Swimming of a Two-Dimensional Flexible Plate at Vanable Forward Speeds m an Inviscid Fluid Journal of Fluid Mechanics VoL 46 pp 337-355 (1971)

2iames, EC., "A Small Perturbation Theory for Cyclóidal Propellers," Ph.D. Thesis, California Institute of Tech-nology (1971).

(4)

of the wing. relative to the flight path are assumed to be small compared to the chord length. Furthermore, when the wing is also thin, it mäkes small disturbances. Accordingly, a hydromechanical development which neglects products of small quantities compared with those that occur linearly will provide first-order estimates of the various physical quantities of interest. . . _.

Physically, the wing 1ás a well-rounded leading edge and a sharp trailing edge Since we are primarily interested in the lifting problem, the thickness is supressed and the wing is treated as a waving membrane. Mathematically, the thickness ofthe leading edge manifests itself in the appearance of a singularity at the leading edge ofthe membrake.

The wing geometry, the flight path, and the transverse motions of the wing relative to the flight path are specified: the problem i to determine the timedependent force, moment, power required to sustain the motion pressure and velocity fields, and energy loss due to shedding The theory is expected to provide useful estimates provided tha wing does nOt cross its own wake.

Associated with a changing circulation about a wing is the shedding of free vorticity at the wing trailing edge. The presence of this vorticity near the wing has a pronounced influence on the lift and thrust produced. The theory inclules the effect of vorticity distributed along a curved path behind the wing..

The framework of the theory admits any flight path along which the wing distrübance remains small. Later on, however, this general situation will be made specific to the simple ease of a wing following a circular arc trajectory (constant curvature) in order to emphasize some salient features of curvature and to demonstraté the analyticàl procedures to be followed for the more difficult case of variable curvature. The results for the constant-curvture case can be readily compared to those for the wing which executes unsteady

niotións relàtivé to a straight-line ph.

K IN EMATI Cs

We consider the prescribed path of the wing described in an inertial coordinate system (, i) which is fIxed with respect to the undisturbed fluid far from the wing. We introduce another coordinate system (x y) whose x axis is tangential to the flight path If this coordinate system has a rectilmear speed V(t)

and an angular speed w(t), each relative to the inertial coordinate system then the curvatureK(t) of the flight path is given by

sc(t) = w(t)/VQ) ,. . . .

(l)

Hence the flightpath is established when the rectilinear and angular speeds are fixed; see Figuie Ï. The motion of the wng is completely described when in addition its transverse motions are given relative to the flight path. This motion is represented by specifying the transverse displacement funçtion of the mean chord line of the wing:

(5)

Hence we use the (z, y) cöordinate system to describe the deviation of the wing away from its flight path;

see Figure 2. .

y

Figure 1 Description ôf Coordinate Systems and the Flight Path (The. motiön of the origin of the (x, y) coordlate, syStem relative to, the mertial reference frame ( 77) traces the flight path. The x axis is taken to bê tangential to the flight path for all tizne asid consequently the rec-tilinear velocity Vft) is along the negative x-axis.)

y

3

Figuré 2 - transverse Displacement e1ative to the Flight Path of a Typical Wing, Section

The wing boundary condition requires that the nOrmal velocity of thé wing relative to the (x, y)

coordinate system - [(a/at) h (x, t)] I I'(y

h)I be equal to the normal velocity [q - (V i- wXx)]. n

of the fluid adjacent to the wing (as seenin the (x y) reference frame) Here n

%7(h(x, t) y)/ V(h

_-

)I is the unit normal vector to the wing,

= Xe + ye is a position iector,

1 2

=

w(t)e

is an axial vector,

K -

V(t) is the rectilinear vélöcity vectOr,

ë values (i = 1, 2, 3) are unit base vectors of the noninertial path coordinate system, and

q =ue +ve

is the absolute velocity vector of a fluid particle..*

(6)

Il' products ot small quantities arc neglected, the boundary conditioh specifies thç vcomponent òf llùid velocity adjacent t the Wing. Consistent with the approximations already mentioned, this velocity

can be specified along th x-axis, giving

la

v(x,v,t) = - + V(t)

) h x,t) +

\at

ax/

y=OtjrI<l ,t>0

The term. ah/at is the nornal velocity of the wing which resúlts when the wing exercises a local time variation away from the flight path. The expression a/at. appearing in this term is a time variation measured in the path frame of refèreice. Wheñ the assumes an attitUde as it glides along the flight path with

speed V. then fluid is pushd laterally by the wing at a rate equal to V ah/. Furthermore, when the

path is curved änd the wing rigid, an additional amount of fluid equal to w(t)x is pushed laterally.

DYNAMICS

In an incompressible flow field devoid of eternal forces and intetnal viscosity, the principle of

con-servation of mass leads to the continuity equation

Çq

0 (3)

The conservation of momeñtum leads to the Euler equation wherein the pressure gradient is balanced by the fluid acceleration

4

*'fl flow field is the region exterior to the wingand its shed vortex sheet.

(2)

q

Y

and 0 (6)

In the subsequent development, it has been found convenient to analyze the problem iii the noniner-tial (x, y) path coordinate system. The form of Equations (3), (5), and (6) remains unchanged when. transformed to the x, y) reference frame. Consequently, we interpret q and to be fUnctions of x and

t and V a/ax.

a = - -

(4)

p

This equatioñ is valid in any inertial reference frame. The absolute acceleration a measures the rate of change of q followiñg a particle, p is the fi iddensity, and p i the local ihsta.ñtaneous pressure. If in

addition, the floÑ fìeld* is rrotational then

q =0

(5)

(7)

M integral of Equation (4) can be Obtained by substituting q = ' . Neglecting the cuadratic term

in q and transforming the localtime variatiOn (as measured in theinertiäl reference frame) to the path coordinate system, we obtain

'a

a'

' _a

a'

+ V(t) )x,y,t)+ ..,(t)

( i-x

at

ax

ay

where 4), thè Prandtl acceleration potential, measures the variation of pressure from the static lèvel

p00 - p (x;y,t)

4)(x,y,t)

-p

Equation. (7) indicates that momentum is impacted to the fluid from local time variations of the wing away from its flight path, from redtiliflear wing motions, and from the effects of path curvatûre or angular velOcities c(t).

Applying the Lplace operator V to Equation (7) aücì using Equations (3) and (5) results in

Ç?2 O

and since both the velocity and acceleration potentials aíe analytic functions of x and y , the complex vanable theory proves useful in the subsequent development Furthermore if one of the potentials can be determined, then Equation (7) can be used: to obtain the other. Mathematically, it is more convenient to solve a boundary value problem for the accelération potential since the pressure is cOntinuoUs acrOss a free vortex sheet (whereas the velocityand hence its potentialhas a discontinuity there) and we will exploit this useful fact. It is then necessary to determine the history dependence (an effect due to vortex shedding) of the boundary data In other words, the presence of the vortex wake transmits a signal back to the wing (at infinite speed iti this incompressible modél) and it is this ififlúence which constitutes the principal difficulty Of the problem.

We introduce the complex acceleration potential F(z, t)

F(z, t) = 4)(x,y,t)+ iW(x,y,t)

and the complex velocity w(2, t

w(z, t) = u (x,y,r) - iv (x,y,t)

of the independent complex variable z = x + iy and the real variable time t. The Cauchy-Riemann equations relate the réai and imaginary parts of each function and the Euler equation relates F(z, t) and w (z, t). Iti complex form the Euler eqúation becomes*

' Operate on Equation (7) with a/ax and i a/ay. Add the two results and use the Cauchy-Riemann equations to obtain the Euler equation in complex form.

(8)

a

fa

a\

a.

F(z, t)( + V(t)

Jw(z, t)-1w(t)

(zw)

az

\òt

öz/

az

l'he arc length

r

alon the flight path is defined as

r=f

V(t1)dr1 ;dr= V(t)dt

With the restrictiôn of positive definite rectilinear speed V(t), both t and

r

increase monotonically and consequently any function ¿f t car be regarded as a function of

r

through the above, transformation. With

this device the Euler equation can be wntten in the following normalized form

a

la

a\

a'

f(z, r)

=

I - + - J w (z,

r) - i g (r) - (z w)

\aT'

òzj

where

F(z,t(r))

f(z, r)'= T and

V(t (r))

Equation (9) can be sOlved for f(z, r)in terms of w(z,

r)

by a direct integration to give

çZ _a

B(z;r)iv(r)f(z,r)I

.

j

ar

where in.the far field we require that 'B(z,

r)

w(z, t)- f(z,

r)-'

O:as z + - and where

B(z, r) = I - iwir)z

Solving for w(z, r) in terms ff(z, r) is accomplished by integrating Eqtiation (9) along characteristic curves whose slopes are given by dz/dr = B(z r) Imposing the initial condition w(z O) = f(z O) = O for

all z on the integration, we btain .

B(z, r) w(zjr) f(z,

= B(z,

T)eiOW

j

B(2,-)[a f(&t) +

where the characteristic çurse is

L .

i = z exp {zLO(r)

- O(] }

+

j

exp {z[Oro)

-

O(e)

_i)

dr0

and 6 (10)

w(t(r))

V(t(r))

}

(9)

Once w(z r) is knOwn-, Equation (10) redus the problem of obtainingf(z, 'r) to one of quadrature whereas Equation (11) serves the dual purpose of-obtaining w(z, r) given f(z, r). We define a complex function of r by evaluating the righthand sidès of Equations (10) and (li), which are equal, at the wing leading edge

z =

1:

¡A (r; K -T a

-

w(z, r) dz

ar

L4(rjc)

= B(1,

r)dO(T)

j

e°

['f(+

iI6)zAf(2M]dA

₍₁₂₎

B(2,)

a - - B(z,) - o

where in this last equation the characteristic curve is given by

exp {iEo(r) ITT _:!'P {1[0ø.o) - O()J}

dro.

In the develópment of the f(z, r) - boundary value problern,,A(r;ic) appears in the boundary data and can be interpreted as a "history effect" since the shedding of vorticity leaves the fluid with a memory.

BOUNDARY VALUE PROBLEM FOR THE COMPLEX

- ACCELERATION POTENTIAL,

The complex acceleration potential f(z r) is an analytic function of z for all time and is continuous everywhère* except ácrôss the wing. Near the ttàiling edge, thè Kútta-hukovskii condition requires that -,

f(z, r) remain bounded whereas at the leading edge, we expect-f (z, r) to be square root singular. - This requirement follows from Equation (10) where we see that f(z r) behaves like the complex velocity w(z, r) near z = 1 The complex velocity is singular at the leading edge in a lineanzed analysis since the assumption of small perturbations is violated. Good lifting qualities are exhibited when the thickness feature of the leading edge is represented by a singularity of-the tquare root type in both the velocity -and pressure fields. The-physical requirement that f(z, r) must decay to zero very 'a.r frOth, the wing and its wake is the remaiñing

oudition necessary to ptovide a unique so1utior

-The solùtion can bè readily fOund by applying the Çaüchy integral theorm to the analytic fuiíction

J(z

l)/(z - l)f(z, r), deforming the contour, and applying the appropriate boundary conditions around

the wing- contour and the -contour at, infinity - One convenient--form of the solutionis - --

-*f(z,T)remains continuous across the free vortex sheet. In fact, by physical requirement3 the pressure is not only continuous across free vorticity but also continuously differentiable That the imaginary part of f(z T)is continuously

(10)

where

and

f(z. r)

=

jA (r;)

i

1/E1+ J

JI

11ff1(E,r)

2

a(r,)

r z+I

in

I

_(Ez)

f1 (,r) = -

B(E,r)v(E,O,r)

-w(z,r) a(r;K) I

S'

fi (E,r)

A(r;,)+

-2 ir

Note that the conditions are satisfied: f(z,

r)

is bounded at the trailing edge, square root singular at the leading edge, and vanishes for z - 00 Also note that the solution is not completely determined until

A (r;)

(or, equivalently, a(r:)) is known. The substitution of Equation (13) into Equation (12) results

in an integral equation for its determination and so in principle the problem is solved.

LEADING EDGE SINGULAR FORCE

Application of the Blasius theorem with a contour surrounding a small neighborhood of the leading edge gives rise to the singular force of thin airfoil theory (here, appropriately modified for the effects of

unsteadiness and curvature). We see from Equations (IO) and (13) that near the leading edge

a(r;K) I

(l+i)

/2(ztl)

8

i:

-

a v(x, O,r) dx

ar

Hence, on using Equation (14) in X5 - ¡Y5 = (ip12)

E. (z,r) dz, we obtain irp

[(a(r;)\21

X---Re

S 2 1+iic irp

r(a(r;K)\21

Y

= -

Im I 2

[\1iic)j

The notationsReand Im designate the real and imaginary parts with respect to the complex unit i = of the space plane. Hence, the singular force in curved flightconsists of a local thrust and a local normal

force.

+O(l)asz-.-1

(14)

i

(13)

}

(15)

(11)

FORCE AND MOMENT

Ïhe resultant force R X + 'Y developed by tha wing is ealily calcïilated in the path coordinate

System. Here X = Xe, Y = Ye2, and both X and Y are positive when R resides in the first quadrant.

An x-directed force X, arises from the component of the pressure difference 4, = p across the wing resolved in the direction of the local forward motiOn, that is

1

X

J

(sp) -!--dx

_ax

and when 4, and (a/ax) h (x t) are positively correlated, X, corresponds to a local thrust. The total x-component of force X is then the stim of X'and the shigiilar force X.

The normal force on the wing and the moment about the midchord are

Y= if (4,)dr'+ Y

(17)

Jx4P)th+YS

whereM is pòsitive in the "nose-up" sense.

The pressure jump bp a'cròss the wing can be expressed as

4p (x, r) =p[Ì'(x, O, t) ci' (x,O, t)]

and from Equations (9) and (13)

where

Ap x, t) = pV(t) Re Ea(r,J

1/ix

Ia

a Re [f1 (, t)] =

_-

+ V (t)

.irp

x=x +

2

r

-{

';

(b'

n= i + Re [a(r; g)] (Io +

- w - b +

After a number of straightforward (but cumbersome) algebraic operations, we obtain

)+b0b1 b0ß1

-(16) (18) , x2

Re[f1(,t)J

L

)

i:

O, t)dx1

(12)

* Equation (20) is derived in' the Appendi,c.

irp Y = Y + irpV(Re [a(r; ic)]

b1)

-irp

M= Y - -

[V(Re(a) + b2)

+ -

(b - b3)]

where the Fourier coefficients b and are defined in Equations (24) and (25). If complex notation is

used for these coefficients, then the real part (with respect to thò imaginary unit / = /Tof the time plane)

of each One must be taken. The dot signifies d/dt.

ENERGY AND POWER

We designate by E the rate at Which ehergy is imparted to the fluid in a unit of time, that is

(q.q)dd

ENTIRE FLUID REGION

where the time derivative is taken in the inertial rèference frame. Using Equation (4) and the divergence theorem, w can cOnvert this expression to an integral aròúnd the wing ectioñ. When tl)e singular leading edge behavior is taken into accotint, we can finally write the räte of energy loss as *

Thus the rate at which energy is communicated to the fluid in a iini of time arises from three sources: The time rate of working of the resultant forces -R V = XV.

'The rate of working of'the moment Mc..x, this is the power required for ä rigid wing to maintain a tangential attitude to the flight path.

The power necessary to execute transverse motions relative to the. flight path is supplied 'by

-

a Ia t (sp) dx; hence it Is the rate of working against the hydrodynanc reaction that opposes

such lateral motions. .

The power input P necessary to sustain the motion is obtained from the conservation of energy

P = - [XV + Mw] .+ . . " (21)

which states that the power is consumed by generating useful work [XV + Mw] while the remainder is utilized in the generation Of a vOrtex wake.

lo

(19)

E

[XV + Mw]

-j1

rl

a I

- h (x, t) (sp) dx

at

(20)

(13)

We now restrict, consideration to a wing executing unsteady motions relative to a circular arc flight

path. For this situation, the rate of change of curvature is zero (i (r)=0), 0(r) = 'nc and Equation (12)

reduces to

.

_a IA (r; ic)

J

ar

f(2, ) d o where 2

=2(r'?)= -

(i_-)exp [ii(r)J

_-An integral equation for the history-dependent terma(r;ic)is obtained by substituting Equation (13) into Equation (22) Consecutively integrating by parts and using the identity

results in

where

CASE OF CONSTANT CURVATURE K

a

Ç2

TLOE-2L

d

v(x,

0,t)

b(t) +

b (t) cos nO ; x = cosO

B(2,)

a

Notice that the b1(t) FoUÎiCr coéfficient contains the influence of the flight path curvature via the c(t) term.

11

(23)

rT

2

(1 IK j

[()+)J d

j)42_i

where b0 and b1 are the first two Fóurièr cosine coefficients of the specified normal velocity v(x,O t, t) f

the wing. Iñ other words we follow Wù1 ¡n expressing v(x, O t) as*

v(x0t)(+V

_)h(xt).w(r)x;IxI<I

I

}

(22)

b (r)

-

'v(x,O,t) cosnOdO ;x= cosO ir

O

Sithilarly the wng displacement function h (x, t) can be expanded as

(14)

(s, g)

=;;;

(s)

-which, when inveEted, givs

i

te+i

a(r;Ic) I

_{2iz j}

h(x,t)= --

_ß0r)+'13(t)

_cosnû;x

cósO Where (25) 2 '

ß(t)

_{= - I}

h(x, t) cosnû dO ; x = cosO

1rj0

The final results can be conveniently expfessed.in teims of both Sets of Fourier coefficients. We designate bS' (s) the Laplace transform of a function br), that is,

b (s)

=

J

eT

b(r) dr ; Re(s) >0

Applying the Laplace transform to Equation (23) reduces the integral equation for a(r ) to the following algebraic equation for (sK)

L

i

Al(s;K)=(\l+_

,b0(s)

-_.- (s;K) - iK) (b0 (s) + b1 (s)) LB1 (s;sc) s; g) e

ds ; e >0

The transforms (s;Ic) and (s;K) in Equation (24) (each depends parametrically on g) are defined by

e' dr

Kctg y)

2 g

+-2 1/2 B1 (s; ) =

J

e

(i

+ i ctg 12 ICT e_ST ctg

_dr

/

_{+ icctg} (26)

These integrals do not have known transforms and must be numerically evaluated for general values of s and IC However for small curvature IC i the ratio A1/B1 can be approximated by the following power series expanson in K

A1 (s, )/B1 (s IC) = H (s) + [2H (s)

- 1] +

j) (2)

K1(s)

where H(s)=

is a ratio of modifled Bessel fUnctions. Using this result in Equation (26) K0 (s)+K1 (s) .

and inverting gives

(15)

+

(b -

b1)}

n= i

irp

- b0 -

b2)

M=Y_

!

Again the real part of each coefficient must be taken when the coefficients _{are complex. The form of} these equations is sittiilar to those of Wu' except for the explicit presence*

_{of w and Y. For certain}

tThere is also an implicit presence of ¡n theb1(t)coefficient as can be seen from E4uation (24).

13 (31)

a r;k

a0 (r)+ ¡Ka,1 (r) + O(K2b) where () -=

f [b()

+

b1 (] Hr

(29)

ai(r)

=

[b(t) -

b0(t)]

Note för c and b (t)of the same order of smallness that a (r,K)has a second-order imaginary contnbution which may be disregarded m this first order analysis _{Because of strong exponential decay, the senes is} expected to provide a useful first approximation for distances r < 27r/K. It should be noted that when the wing executes unsteady motions relative toa straight line path (ic =O), then Equation (29) appropnately reduces to the comparable result foUnd by WU'

In this case, the singular force at the nose (Equation (15)) becomes

xs [a0(r) + a0* j2 +OQc2 b2).

(30) Y .2icX +

0 (g2)

The asterisk denotes the complex coñjugate With respect to /

= /ï

_{the complex uriit of. the time plane.} Furthermore, Y is a third-order quantity of smallness unless the unsteady motions produce large accelera-tions about the nose of the Wing hi this case wifi become large.

Equations (19) become

+bö ß0)(a0 b1

+ß1)+ß0ß1+ w(a0 +172)

(16)

unsteady transverse moti9ns when the fluid acceleration about the leading edge is large, the singùlar force correspondingly becomes large When this situation prevails, X-Xi and become influential m the

Y and M expressions. Hwever, under the optimum condition of minimum energy loss while maiiitainiiig a specified coñstant thrúst, the singular local thrust was found to be small.3

CONSTANT FORWARD SPEED V ABOUT A CIRCULAR ARC TRMECTORY

Here we treat the s.mple case of an uncambered wing traversing a circular arc flight path of curvature c which impuhive1y accelerates from rest to a constant local forward speed V; lt is further assumed that the wingmaintains a constant angle of incidence a to the flight path fór äll time. Hence, by' Eqüatiàn :

(24) '

= 2 Va, t >0

b1=Vg,t>O

(32)

b

0;n2 ;t>0

The singular lift, Y can safely be neglcted bere since it is O( provided iç and a are comparable in magnitude and small compared to unity Equations (29) (31) and (32) then give

Y=-7rpV2 [2a+] W(r)

M=_V2.[K_(2a +K)W(r)]

The time-dependent function W(r) is the Wagner functiòn which gives the growth of lift and momént. lt can bé approximated witl an error no larger than 2 percent over the entire r range by

/

2''\

W(r) _I

H(rr)d'._( i

(34)

J0

\

4+r,

We note that Y'. O when a - g/2 and that the moment about the mìdchord becomes independent of

time. As expected, the effect of curvature manifests itself as an effective constant wing camber since the curved flight path enables the flat Wing to get a better grip on the fluid.

3Wu, T.Y., "Hydromechäiics of Swinming Propulsion, Part IL Some Optimum Shape Problems," Journal of Flüid Mechanics, Vol. 46, pp. 521-544 (1971).

-'Lighthill, M.J., "Acquatic Animal Propulsion of High Hydromechanical Efficiency," Journal of Fluid Mechanics, VoL 44, pp 265-301 (1970).

14

(17)

TIME HARMONIC MOTION WITH CONSTANT V AND CONSTANT ic

In this section we want to supplement the Wu treatment of harmonic time motion' by including the influence of flight path curvature ic. The transverse wing motion is specified by

h(x,t)h1(x)e/7t

_;xI<l

_,t>O

₍₃₅₎

where y is the radian frequency of the harmonic motiOn. The prescribed normal velocity across the' wing is

v(x,O,t)=VKx+v,(x)e17t.;Ll<l;t>O

v(x)

v(-- +/o)hi(x)

;a=7/V

₍₃₆₎

y (x, O, r) = O ; t O

and since the curvature term in Equation (36) does not involve the time harmoñic motion, its influence cari be treated separately. We express v1(x»7t

v1(x)e7t =

_c(t)

+'

_{c(t) cos nO ; x = cos O}

2 çn

c(t) = ejTt

_J

ir0

v1(x) cos nO dO ; n = 0,1,2,..

a0(r) n [c1(t) + Vsc] - [c0(t)+ c1(t) + Vic]

For large r we çan develop an asymptotic expression for Equation (38) which reads

a0(r) = c1(t) - 9(o) [c0(t) + c1 (t)] + -

+ O(r2) ; r'» I

; r= Vt

K1 (jo)

K0(jú)+K1(ja)

f(ci)+f

(a) ;o

7/V

9(ú) is the Theodorsen function and o is the reduced frequency based on the normalized semichòrd length. We see from Equation (39) that the éffect of the constantly curved, flight path diminishes in versely with the distance traveled by the wing.

For small r, Equation (38) becomes

15 {

jI.

H(s)ds

exp ['r(sjo)]

(sjo)

(38)

E-r

5

+---r2+O(r3) ;r1

₄ ₁₂₈ (37)

(39)

(40) From Equation (29)

a(r)a V[l

W(r) ] +c1(r) - [c0(t) + c1(t)] 2iri

(18)

From Equations (24), (25j, and (37):

b(t)c(t) forn=O,2,3,4,..

b1(r) VK

c1(t) n e/lt v1x) cosO dO ;x cosO

¿3(t)= eilt

if1T

7ro

16

and with the results of Equation (38) to (41) we can directly obtain estimates for the thrust, lift, and moment production by using Equation (31) for the csse of harmonic time motiön.

CONCLUDING REMARKS

Within the framework of the theory a wing havmg variable local forward speed V(t) along a curved flight path can be treated L The wing can exhibit arbitrary transverse deformation or displacements relative to its flight path, so long äs these are admissible in a linearized theory. The details for the case of con-stant nonzero curvature have been presented in the preceding section.

The principal difficulty in the unsteady wing problem is that of determimng the history effect düe tO the vortex shedding. l'his effect can enormously influence the force and moment produced by the wing. We have demonstráted here that some analytical progress can be achieved for constant curvature by simphfymg the integral equation for the time history terms A(r,x) and a(r K) For variable path curvatùre,however, it appears that the integral equation defined by Eqúations (12) and (13) must be handled numerically.

Von Karman and"Sears5 have shown that the vortex wák nearest the wing has à pronounced effect on the lift and moment whereas more distal vorticity has a greatly diminished influence. 11th suggests that an estimate of the shedding effect for nonciverlapping but variable flight paths can then be obtained by iterating the integral euation, Equation (12), with its zero curvature solution, Equation (29) with

Further analysis of the influence of vortex shedding seems possible for the case of small curvature with rapid oscillations In this situation (r) > (r)

For an overlapping flight path the wmg perceives a gust situation when it passes through its own wake. The history effect is again present in Equation (12) but now the influence of:vorticity shed at an earlier time can become qute signiflcant at a much later time. .Much work remains befOre the fluid mechanics of this interesting problem can be quantitatively predicted with reasonable confidence.

5Von Kannan and Sears,i "Airfoil Theory for Nonuniform Motion," Journal of Aeronautical Science, Vol. 5, pp. 379-390 (1938).

(19)

ACKNOWLEDGMENT

The author is grateful for the guidance of Professor Theodore Yao-tsu Wu in the preparation of this work and for the availability of advance copies of lus reports on swimming propulsion1 which served as a model for this problem.

(20)

APPENDIX

DERIVATION OF THE ENERGY BALANCE EQUATION

is the rate at which energy is communicated to the fluid in a unit of time, thát is

.

d1

E

-j

p(q .q)dd

dt'

. 2

-

-ENTIRE FLUID REGION

where the total time derivative taken with respect to the inertial coordinate system (d/dt') operates On the kinetic energy of the fluid. This expression can be written as

fp.a)dd

(2)'

EN'r1RE FLUID REGION where a. the. absolute fluid acceleration, takes the simple form

a a..=

since Vx = O in the fluid region. From Equations (4) and (8), the ÉÚler equation reads

a =

and therefore

(.a)V(qCF) since

q=O

(3)' Substituting Equation (3)' nto Equation (2)' and using the two=dimensionál divergence theorem gives

pcI(q.n)d2,

(4)'

where the line integral is taken around a counterclockwise .cóntour C which entirely encompasses the fluid region. The unit normal vector n is the outward normal to the fluid; see Figure 3.

In terms of integrations over the particular arcs of C, Equation (3)' becomes

Ê=Ç

P(.P)d+j p[Vq.ñ)q.n)Jd

cv+

f

r'

+1) P4)(.Pi)dZ+j

p[(q.n)'(qan)d&

CE, '-'w 19

(21)

s

e.! ..

V .

cv +

cv

-'Figure 3 - Contour C Encompasing the Fluid Region (It consists of arcs C,

c7

beneath and above the vortex

ke, C , and C beneath and above the wing,. C a cixde of radius E suìrounding th leading edge, and C,,, a 'circle of infmite radius The direction of the oùtward

normal is shown on C) The properties of the ititegrand of E are as follows:

(q . n) is continuous acrnss all surfaces in the fluid; itiore specifically, across the free vortex

wake and the bound vorticity which represents the wing, (q . nr = (q

'I is continuous everywhere except across the wing where we expect a pressure jump.

3.. and q are singular at the leading edge and consequeñtly we anticipate a contribution from hè integral about C6.

4. (J) and q decay sufficiently fást away from the wing and its vortex wake. From Equation (13), 'J)

O(z) as

and from Equation (9), q _..O(z) as IzI oo Invoking property (4)

pcJ)(q

_{. n)d = O .}

oki property (2):

J

p[q , nr

(q . J d O.. Since ip(x r) = p ['J)(x, O, t) x, O, r)i, we. can write E as

In the first integral abov (q . nr = v (x, 0, t) (the minus sign arises because the unit outward normal to the fluid along the topside ofthe wing has components n = (0, 1)).

Considering the integral about the leading edge

(j5 p (J)

(q . n) dR,, we let z + 1 = ee on C6

20

(22)

so that d2 &O and Ô <O <27r. Then th

unit outward normal to the fluid on C has components

n = ( côs O, sin O). From Equation (14) the components of q = ù, y) are

u Re [Ae 10 j s,/

+ 0(1); E

O

y

-

Im[Áe'j j

0(1) ; e

O

where

AA

21 Furthermore, from Equations (9), (iO), and (14)

Re[a(r;ic e_tOh'2]

_+O(1)ase0

so that on C

n)d=a VÉe[ä(r.,,c)e_2] Re[Ae!2]

de

and therefore

+w[.2ARAf]

Using Equation (15) gives

p(q.n)d9VX5+cY

cc

Hence Equation (5)' becomes

jv(x,0,.t)/p(x,r)dx + VIS

+

Substituting Equapns (2), if6) and (17) into Equation (7)' ves C'

òh(xt)

EVX+cM

J

a

p(x,t)dx

(23)

INITIAL L)ISIR.ibUIION

ÑEPOT

409d

SERIAL

1

WES/LIbNAÑY

2 ChUNk 's3 3

ONR BOSIUN

4

ONN ChILA&O

5

ONN PASAULNA

6

NNLdbe7' Lih

7

NHL ORLANDO

8 (JSNA Lib 9 NAV-'b5COL Lib

10

NROTC

NAVAuMii'U. MIT

11

NAVWARLUL

12

SHIIS ¿052

13

SHIPS UJ412/PETLaçON

14

SHiPS uJiL

FAC 032C

16

ORO 03/SEIUhAN

.17

NAVUCLANU 1640

18

_{NAVAIRULVCLN AOl}

_Ib 19

NELC Lib

20

_{NAVVS.ACEN 1311 Lib}

21 NAVUSLACtN bOOS/P ThbULA

22

NAVUSACEN 13111

ib 23

NAVUSE.ACLN 2SO1/HYT

24 NAVPNSCLÌ'I 7,3 L1, 2

CIVENLAb Li! Lk

2ó

NUL 730 LIb

27

NWL Lib

28

NP[LAb NUSC/LIb

NLONLAI NUSC

. 30 NAVSHIPYL) bRLM/Llp 31. NAVsf-I1PYI) bSN/LiU 3

NAVSHIPYD CHASN/Ljb

33 .

NAVSHIPYU HUNÌLRS

i/Llt

34 NAVSH!PYU Lbt.ACH/. 1h 35

NAVSHIPYU MANL/L1t

3b

NASH1PYD MANL 2

37 NAVsHIh-Yu PEANL/L1b 38 NAVSHIPYI) PIlILA U 39

NAVSHIPYU PTSMH/Ljb

40 SEC 603'+d 41 SEC

bill)

42 SEC b11'+rt 43

SEC oi0

44

SLC bub

45 .

SEC bi4ub/FUNCANNnN

47 SEC b1+ 4b AFUSN/IILLH DIV 49

AFFDL/S/uLSIN

50 1)LiC 22

(24)

SERI AL 51 DOC 52 '

DUC

53 DDC

.54

DDC 55 DOC

56:

DOC 57 DOC 58 DOC DDC 60 DOC 61 DDC 62 BUST'ANU/KLEBANOFF 63

LC/SCL

TECH' DIV 6 ' 'MMAL.L

65 MMA/MANITIME RES i-EN

66

NSF ENN 01V LIB

67

DOTLIB

68

U 8RIDGEPORT/URAM

69 U CAL rSLRKELEY/DEpT ÑAME

io

u CÀL"NAMÈ/PAULLIÑG

71

u CAL NAML/WEBSTL

72

U CAL NAME/WEHAUSEN

73 "WCAL' SAN. DIEG0/, LIS

74

U CAL SCRIPPS LIb

75

CIT AEÑO LIB

76 '

CIT/ACOSTA

CIT/WU

78 CATHOLIC U/CASARELLA

79

CATHOLiC U/HELLLN

80

COLQRAÛÜ' STATE U FNGR RES CEN

81

U CONNC'i ICUT/SÇOTTNON

82 'COÑNELL U/SEARS

83

FLUNIuA ATLANTIC ii 0E LIb

84 GEORGE WASHINGTON U/F lEp

85 HARVARD U MCKAY 'Lib

86 '

'

HARVARD U/cANIEN

87 U HAWAII/bNEÏSCfr$NFIOEN

88

U LLLINUIS/NOBENIÇON

89 '

U IOWA INST HYD

pES L'ÌB

90

u 'IOWA iH/KEÑNEUy

91 ' U iQA It-Ik/LANOWERER

92 jU,-iÑs t-IUPKINS U/tHILLIPS

93 U 1SANSAS CIV ENGN LIB

KANSAS SlU ENGR FAP/NESMITH'

95 '

LON( ISLAND U/PN1'E

96 MIT ÙCEAN 'ENGR L1i

MIT 'OCEAN ENGR/Ab,OwITZ

98 MIT OCEAN' ENGR/LEFÑEY

99 ' MIE OCEAN ENGN/MAriOEL

100 ' ',

MIT OCEAN ENGN/NEMMAN'

(25)

SERIAL 101 102 103 104 105 106 107 lOA 109 110 Ill 112 113 11'. 115 Lib 117 118 119 120 121 122 123 124 125 126

127 i 2

129 130 131 1 3 133 134 I 35 136 137 1 3b 139 140 141 i '+2 143 144 145 I 4b 147 148 149 ISO

MIT PASUNS LAt3/IpPN

U MICHIL,AN NAME LTB

U MICHIUAN NAML/CnUCH

U MICHIUAN NAME/HMMIiT

u M1CH1i,AN NAML/UILV1E

U MICHIGAN WILLOW NUN LAbS

U MINNr.SUTA SAFHL/P¼1LLLN

U M1NNtO1A SAh-iL,'SCHiEbL

U M1NNE.SUFA SAFHL/SONG

U MINNLSOIA SAHL/WE1LEL

NOINE UMML ENGR Lib

NOIRE UAME/STNANUHAuEN

NEW YURis U/FIENON

NYU COURANT INST/pETERS

NYU COURANT INST/çTOKEN

PENN STAlE U ARL

t Ib

PRINCETON U/MELLO.

ST JOrINS U/LURYE

SWRÌ APL1ED MECH REViEW

SW Ni/A

b

N A M SON

STANFORD U CIV EN(,N LIB SÌANFURI) U/SÏRE.l

STANFORD RES INSI Lib

S1EVENS INST UAVII)SÜN LAb/

SIT DAVIDSON LAU/pRESLIN

SIT L)AVIUSON LAB/TSAKONAS

IEXAS U ARL Lib

UTAH 5TAIE U/JEPPcON

VPI/MOOk

VPI'NAYFEY

U WASHII\IL,TON AI-IL Ib

Wttffl

INST/LEWIS

WEbti INS 1/WAND

WHUJ OLEAN ENGR o-T

WPI ALDEN HYOR LAR Lib

SN A ME.

AEROJETGENERAL/NEGG IT T

HEFHLt.f-1E.M STEEL NFW YÙNtc/LIts

8E.THLL.I-IEM STEEL StANNOWS

bOLt BENANEK AND NEWMAN Lib

BOLl tiLNANEK .

NE'MAN/JACKSUN

CAI'4bkIU'E ACOUS/Ji iNGEN

CORNELL AENO LAB/pITTEN

EASTERN RES GRoUp

ESSO uE

DIV

GEi DYN ELE.0 bOA I/t3OATWNIGHT

GIBbS

CDX

ÇYL)NONAUIICS Lib

LOCKHEED MS/wAIU

L)UUGLAS AÌNCRAFÌ/kESS

(26)

SERIAL

151 152 1.53 i 54 155 156 157 158

DOUGLAS AIRCNAFT/ÇMITH

NEWPOÑT NEWS SHIPpUILOIN(, L1

NIELSEN ENGR/SPANç,LEÑ NA

SPACE/UJIHANA

OCEANIC S/SAPLAN

SPEkRY SYS MGMT

_Lib

ROBERI IAGGANT

TRACOR 25 SL+IAL

CENTEN UISTNIbUI[ON

159 C 1500

CUMMINS WiLLIAM E

loo C 15U

STUNIL

EURGE k JN

161 C i:seo

WENMÌER RAYMOND

i 6g C 1540

MONGAN WILLIAM B

I 63 C 1548

OX BUCE D

16'.

C 15

SALVESEN NILS

11)5 C 15b

WILSON MICHAEL B

166 C 151)0 HAULEN ,jACUUES B 167 C 151)'. JAMES EDWIN C. 168 C 151)'.

JAMES EDWIN C

169 C J51)'. JAMES E0WIN C.

170 C I5b4

JAMES EDWIN C

171 C 151)4

JAMES EDWIN C

1.72 C I5o+ LEE ChON(ì MOOPS

173. C 151)'.

CURPHLY RiCrAND M

.114 C 151)4

FELDMAN JEROME P

ils C

1564

wOLF..KA!PTN K

(27)

I NOV 65 I'I ¡ S/N. 0101.807.6801 UNCLASSIFIED Security Ciassili UNCLASSiFIED Security ClassiÍicjtj

--" DOCLJMEP4T CONTROL DATA - R & D

S c-ut ty class,(tcat or, of tillo body of abstract n a ndex, ,g annotai n n u i it entered when Ile overall report Is elasst fled) I. ORIGINA TINO ACTIVITY (Córporate author)

Naval Ship Research and Development Center Bethesda, Maryland 20034

-

--20. REPORT SECURITY CLASSIFICATION

UNCLASSIFIED

2b.GROUP

- ".

. REPORT TITLE

A LINEARIZED ThEORY FOR ThE UNSTEADY MOTIONS OF A WING IN CURVED FUGHT

3

-A-DESCRIPTIVE NOTES (Type of report nd inclusive dates)

5. AU THORISI (First name, middle initial, last nime) - -

-Edwin C. James

6. REPORT DATE

-August 1973

7a TOTAL NO. OF PAGES

27

7b. ÑO. OF REFÌ

5

Sa. CONTRACTOR GRANT NO.

-b. PROJECT NO, ,

Program Element 61 152N Task Area ZR0230101

a. Work Unit 1-1564--001

90. ORGINATÖRS REPORT NUMBER(S) - -

-4098

Sb. OTHER REPORT NO(S) (Any-other numbers that may be aìsigned

1h, s report)

IO, DISTRIBUTION STATEMEÑT

-Approved for Public Release: Distribution Unlimited

II. SUPPLEMENTARY NOTES _{12. SPONSORING MILI TAY 'ACrI VITY}

In-House lED Program

-

-13. ABSTRACT . ' - -

-A linearized theory is developed to treat small-amplitudè unsteady motions of a wing in curved flight at variable local forward speeds in an inviscid incompressible fluid. The wing geometry motions and flight path are specified and the problem is to obtam the time dependent force moment power required to sustam the motion pressure and velocity fields and energy loss due to the shedding of vorticity. The theory is expected to provide useful estimates provided the wing does not cross its own wake.

The effect, of path curvature is a particular interest in this investigation. To contrast this effect, the results can be readily compared with those for a wing in straight-line flight.

.-

--

(28)

(PAGE 2)

UNCLASSI FIFfl

Security Classification

DDFORM 1473 (BACK)

I NOV 65 UNCLASSIFIED

Security Classification

KV WORDS LINK A LINK B LINK C

ROLE WY ROLE WY ROLE CT

Wing - 2 Dimensional Unsteady Linearized

Large Amplitude Trajectory Lift. Thrust, Moment

Energy Loss C'

Vortex Shedding Power In-Put Pressure Field Velocity Field