Chaplygin Theory: Determination of Potential Flow and Forces in Fluid Dynamics, Study notes of Acting

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NATIONALADVISORYCOMMIITEE

# FOR AERONAUTICS

TECHNICAL MEMORANDUM

No. 1156

ON THE TBEORY OF THE UNSTEADYMOTION OF AN AIRFOIL

By L. I. Sedw

Central Aero-~drodynamical Institute

Washington

July 1947

I

lllllllllllll

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL hmmrwam

ON THE THEORY OF

The paper presents a

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NO} 1156

THE UNSTEADY MOTION OF AN AIRFOIL*

By L. I. Sedov

systematical analysis of the problem of the determination of the unsteady motion about ail airfoil moving in an infinite fluid that contains a system of vortices and the deter- mination of the hydrodynmical forces acting on the airfoil. The hydrodynamical problem is reduced to the determination of the function f ({) which transforms conformably the external region of the airfoil into the interior of a circle. (^) The proposed methods of determining the irrotations,lmotion of a fluid that is produced by any motion of the airfoil.are especially simple and effective if the function f (~) is rational. (^) .% an example the flow is deter- mined for the case of an arbitrary motion of an airfoil of the Jouliovskytypo. (^) The l“ormulasobtained for the determination of the hydrodynamical forces by means of contour integration are similar to those given by S. Chaplygln, These formulas are used to deter- mine the force acting on the aizi’oilin the cases where the unsteady motion is potential throughout and the circulation about the airfoil is constant and also when the fluid contains a system of vortices. A full di~cussion is given of the concept of virtual masses together with practical formulas for computing the virtual maqs coefficients. A table is added .givin&the virtual mass coefficients for different types of Joukowsky airfoils. (^) For the case uf motion with constant circulation the followin~ theorems were obtained. (^) Every airfoj. possesses a fixed point such that the force, depending on the cir- culation, for any motion of the airfoil is determined by Joukowskyfs law in terms of the velocity of this -point,the moment of the hydro- dynamical forces about this point being independent of the circula- tion. (^) Formulas are also given for the dete~mination of the forces and moments acting on a thin, slightly curved airfoil of’Joukowsky profile for any motion with constant circulation. For the case where the circulation about the airfoil is zero the condition that the fluid velocity at the sharp edgo of the airfoil should be finite for a continuously potential flow is reduced to the requirement that the traveling polhode should be a fixed straigh~ line, perpendicular to the first axis for every airfoil. (^) We have “thesimilar geometrical condition for the case where the angular velocity of the airfoil and the circulation are independent of the tflme. ?/ehave also given fozmnzlasfor the computation of the hydrodyn~~.cal forces when the flow contains any system of vortices. (^) These forces depend.on the

*Central Aero-Hydrodynamical Institute

NACA TMNo. 1156 3

Starting from flows with surfaces of discontinuity within in fluid L. Prandtl developed theoretically on an expe~imental basis the approximate theory of a wing of finite span for-steady motion (reference 1). He also gave examples of the accurate solution of certain two-dimensional problems on the unsteady motion of an ideal fluid with a line of &iscontinuity In the form of a spiral (refer- ence 2).

Birnbaum set up an approximate theory for unsteady motion, the body being replaced by a system of bound vortices (referenceF3z, 4).

In studying the unsteady motion with surfaces of velocity dis- continuity a fundamental difficulty lies in the fact that the mechan- ical characteristics of the body-fluid system depends not only on the condition of motion of the body at a given instant of time but also on the preceding motion of the body. The motion of the system at a given instant of time is thus a function of a combination of all preceding states of motion of the body. (^) In particular, the exact solution of the two-dimensional problem of unsteady notion, with lines of velocity discontinuity starting out from the trailing edge, encounters very great mathematical difficulties and as yet no solu- tion has been obtained. (^) This problem was characterized by Prandtl as “hopeless” of solution.

The forces acting in a two-dimensional flow on a plate from the trailing edge of which a line of velocity discontinuity is given off were investigated by Wagner (reference 5), Glauert (reference 6) and Keldish and Lavrentev (reference 7). (^) In these investigations it is assumed that the llno of discontinuity coincides with the straight line which is a prolongation of the plate. This assumption IS associated with another, namely, that the angle of attack of the plate remains ird?initesimallysmall.*

In the general case it is convenient to isolate from the hydro- dyneraicforce~ those which depend only on the velocities and accel- erations of the body. (^) These forces agree with the total force for the motion of the body in an itiinite fluid on the assumption that the motion of the fluid is continuously potential. The determination

*In the given case by angle of attack is meant the angle between the velocity of the center of the plate and the direction of the plate.

4 NACA TM No. 1156

of the motion of the fluid and the forces in this case is a classi- cal problem (reference 6), The additions to these forces thus isolated will depend essentially on the position and motion of the singularities of the flow outside the kody.*

By the akove approach the study of the continuously potential flows enters the study of discontinuously potential motion of an ideal fluid as a component part.

The general theorem of the forces independent of the singulari- ties of the flow was worked out by Thomson and Tait (reference 9) and Kirchhoff (reference 10) as far tack as the middle of the last centwy. The theory of =rchhoff was generalized.by S, A, Chaplygin (refer- ence 11) in the case of the two-dlmenstonal problem to the mot:on of a wing with constant circulation. In this work methods were given by Chaplygin for the determination of the potential flow of a fluid and the computation of the forces based on tinetheory of functions of a complex variable, The present paper gives a systematic presenta- tion of the theory of unsteady motions for the case of Chaplygin and the case where a number of isolated vortices exi~t within the fluid. There are a large number 01’unsteady motions, the ch%racter of which will be clear from what follows, for which these theorie~ may have a direct practical value. Moreoverj as has already been pointed out, the results of these theories may enter as componenc parts of inves- tigations dealing with the most general case of the mo~ion of a fluid.

2, FUNDAMENTAL RELATIONS

We consider an arbitrary plane-parailel motion of a cylindrical wing within em incompressible fluid such that the plane of the motion is at right angles to the generators of the wing, All mechanical characteristics will be computed with the aid of a Cartesian system of coordinates xOy^ fixed invariably to the wing,^ For convenience the vectors will be considered as complex numbers. Yor example the position of a point will be detemined by the vector

z = x-!-iy i. 61- -— -. —

*These singularities may be vortices, boundaries of the fluid, surface of discontinuity, etc.

.—.———-. ., , , ., ,,,, ,,,,, ,

(^6) NACA ‘2MNo, 1.

while the f’unctionethemselves are connected ly the Cauchy-Rienann

so that

whence

cp+ i$ = w(z)

A F=u - iv =

w(z) is known as the comylex potential

ail

-F

dw dz

function.

For points of the region of potential motion the pressure p(x, y, t) may be computed with the aid of the Lagrange integral, In the absence oi external mass forces we shall have

&~(”’+‘2)

P~(@-P~

where p is the density of the fluid, (^) PO (t) is a fuilctio~ldepend-

ing only on the the. (^) This function is detemnined by the given pres- sure at any one point of the fluid. In the above formula of Lagrange the partial derivative - w/at is taken on the assumption that the potential cp is expressed as a function of’the the and the coor-

dinates f,17 defined in the stationary system of axes. Between

the coordinates ~,11 ad the coordinates x, y in the moving system we have the relation:

f = C(X, Y, t); q = 7(X, Y, t)

Evidently @t and aV/bt are the projections on the sta- tionary

We

axes of the velocity of “thetransporting motion,

thus have

a~ (x,y, t) acp(f^ p)+mt+w~

at = ------%- ‘“– a~ at

or

NACA TMNo. 1156 7

where (^) ~ is the velocity of the transporting motion.

Making use of relation (3) we rewrite the integral of Lag.’ange for the case where the potential Is determined as a function of the time and coordinates of the moving system:

P= po(t)-p~-;^ (U2 + V2) +^ p^ (Uu ‘tV-V)^ (4)

Since in what follows we shall throughout make use of the mov- ing system of coordinates we shall always compute the pressures by the above formula.

In d.ifferenti.stingvectors with respect to time we shall dis- tinguish derivatives with respect to the moving and stationary sys- tems of coordinates. The first we shall denote by the symbol (^) slat and the second by 5/5t. The former are used because the vectors under consideration depend also on the points of space. These derivates are connected, as is known, by the relation

which on replacing the vector = %y the complex number a beoomes

3. FORMULAS FOR T!KEFORCE AND MOMENT EXERTED

BY THE FLUID ON AWING FOR UNSTEADY MOTION

In this section we shall consider the general formulas express- ing the total hydrodynamic force and the moment of hydrodynamic forces with respect to the origin of coordinates by means of inte- grals, taken around the coirbo~r of the wing, of functions of a com- plex variable. With respect to the velocity field of the fluid we assume only that in the vicinity of the contour of the wing C the field has the potential ~(P,t) where P is a point within the fluid and t the time. (^) At the traillng sharp edge of the wing in

1

NACA

With

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TM No, 1156

the aid of (2) and (3) we reduce relation (1) to the form:

(dX+idY) = ~ idz + i~dC~- “(z %!!p :[FZ +

Integrating thie equation we obtain the formula for the total force on a unit width of wing:

In the above equation (^) r denotes the cjrculation taken on the con- tour of the wing counterclockwise and ZM is the coordinate of the starting point of the line of velocity discontinuity.

Remembering that along C~. Uoy - V@ - ~^ (#+y2)^ + const.

we compute the integral $ zd~. (^) Integrating by parts we obtain c

[ 1

j Zd~ =-$ TJoy - VOX - ; (X2+Y2)((j,-x+idy) c

l?ut

$xdx=$ydy =$x2dx=$y2dy=

Thus we may write

$ Zdl+f.- (^) U. $ ydx + tV# xdy + c

The formula of Green, applied to the

; $ y2dx + ix2dy

present case, gives

-$ydx=ixdy =~sdxdy=s$ y2dx. -2ff ytjXdy.

2 Sy* h x2dy = 2 JJ xdxdy = 2 SX*

where S is the area bounded by the wing contsur, x* and y*

10 NACA TM No. 1156

are the coordinates of the center of gravity of the wing area. Hence

[ 1

$ zd$= Uo+iVo+W(-y++ ix*) S.Sq~ (6) c

The magnitude here denoted %y q* is evidently no^ other than the velocity of the center of gravity of the wing area. Substitut- ing the values of the computed integral in formula (5) we obtain the general formula for the force exerted by an unsteady flow on a wing

(I)

The above equation is analogous to the Elasius-Chaplygin equation for Bteady flow. The integral in the equatjon may be taken about any contour L enclosing the wing if between L and C there are no singular points of the complex potential function. This fact makee equation (I) suitable for the computation of the acting forces.

If: 1) the wing moves forward with constant velocity 5q*/5t = w. o

2. the circulation about the wing is constant r = const, dI’/dt. 0

3. tie fluid is infinite, over a finite distance there are no vortices and the fluid is at rest at infinity, then the last term in equation (I) is equal to zero since in this case the integral does not depend on the the. Since in this case dw/dz outside the wing is throughout homomorphic andah infinity i~ of the order l/z, (dw/dz)2 is at infinity of the order l/z2 and therefore the first integral likewise vanishes. Thus the acting force is reduced to the Jourkowsky force

be

x+iY. (^) fPqor

If there are external mass forces an Archimedes force is to added on the right of formula (1).

Evidently in may be taken over and C there are tion.

If the fluid

NACA TMNo. 1156

loth formulas (I) and (11) the two first inteGrals any contour L enclosing the wing^ if between^ L no singular points of the complex potential func-

is infinite and there are only isolated singu- larities within the flow (sources, vortices, dipoles, etc.) both integrals in equation (I) and the two first integrals in equation may be determined as the residues about the singularities of the

(II)

functions under the integral signs. The last i~tegral in equation (II) in this case may also be computed with the aid of residues if it is possible to construct an analytical function of the complex variable z having isolated singularities outside C and assuming on the contour C itself the walues :.

From what follows it will be seen that the presence of isolated singularities in the flow, if the transformation of the outside region of the wing into the interior of a circle is effected by a rational function~ the last integral in equation (Ii) may be obtained as a sum of residues.

4-, DEI!ERMINATIONOF POTENTIAL FLOWS

A solution of the problem of determining the potential flow of an infinite fluid for any motion of the ‘~ingwith the aid of a function conformably metippif.gthe externai reglm of the wing on the upper half-plane has previously been given by S. Chaplygin. In this section we shall give methods of solvlng this problem with the aid of the function z = f(~) that Transforms the outer region of the wing in the z-plane into the interior of a unit circle K in the ~-plane.

For definiteness we shall assume that the center of the circle K coincides with the origin of coordinates and that for z = ~, ~ = 0. The function f(~)^ near the origin of coordinates has the following form

z =f(~) = :+ko+k1~+k2c2+,.. (1)

where P(c) =ko+k1~+k2~ +... homomorphic everywhere within K,

NACATM NO. 1156 1s

We denote the complex potential function of the flow under consideration by W. (z) =CPo + i$O where cpO (x, y, t) and $0 (x, y, t) are single-valued and harmonic functions ever~here outside the wing. To determine WO (z) we have on the wing con- tour the condition

We represent (^) Wo (z) in the form

W() (z) = Uowl (z)-!-vow~ (z) -I-(IJW3(Z)

so that W1 (z) W2 (z) W3 (z) are homomorphic everywhere out- sld.ethe wing and their imaginary parts $1, $ 2, *3 on the con- tour of the wing C satisfy the following conditions:

$1= Y; W2=-X;W3=-; (X2+Y2) (3)

It is readily seen that WI (z) is a potential function of the potential flow for forward motion of the wing along the x-axis with unit velocity, (^) W2 (z) corresponds to the forward motion of the wing with unit velocity along the y-axis and W3(Z) gives the potential flow of the fluid in rotating the wing about the origin of coordinates with an angular velocity equal to unity. The func- tions w~ (z), W2 (z), w3 (z) are determined by the geometric properties of the wing contour.

Replacing in (^) W(-J(z)) z by ~ we obtain a function (^) W()(c) homomorphic within K. In a similar manner we obtain the func- tions W1 (z), W2 (z), W3 (z) homomorphic within K. To deter- mine these functions we write boundary condition (3) in the ~ -plane on the contour of the circle K in the following form:

Imag W2 = Imag - if(t) (^) (4)

NACATM NO. 1156 15

fl (~) and f2 (~) in euch a m~ner that fl (~) is hol~orphfc

everywhere within and on K, (^) an? f2 (~) everywhere outside and on K and that

w3(~) = Zfl (t)

Thus for ICI = 1, -;f(~)F (^) () 1 c,

=fl (g) +fz (~) is purely imaginary, hence on the circle K

()

l\

Realfl (~) = -Real.f2 (~) = -Real= ~

It is evident also that on K

(lo)

The functions (^) fl (~) ~d -~ (1/~) are homomorphic within and on K, and accordir.gto (9), on the circle their real parts are equal. Hence throughout the ~-plane the equation holds: /\

where q is a purely imaginary constant,

By virtue of relations (10) Itl =

Imagfl (~) =

Hence the function 2fl (~) - q

and (11) it is evident that for

Imagf2 ({) + q

is homomorphic within K and for Itl =1 thetiaginmy part is equalto -i(x2+y2)/2 andthere- fore with an accuracy up to an unessential additive constant rela- tion (8) holds. The above splltting of the function is particularly easy to carry out in the case where f(~) is a rational function.

Evidently W3 (~) may be determined also with the aid of the Schwarz integral (reference 12). Since the real part of the function iw3 (~) on the circle K is equal to 1/2 f (~) 7 {1/~)

where C is a purely

Transforming, we

imaginary constant which

obtain

NACA TM No. 1156

may be neglected.

If f(~) is single valued and has only an isolated singu- larity the integral in equation (81) may be computed with the aid of residues. Setting

W3(C) = Clt + c~c2 -tc3t3 +.. ,

With the ald of equation (81) for the coefficient c1 we obtain the formula

The series (811) first expand in a

$(e)= -~f(eie)

for w3(~) may readily be written out if we Fourier series the function

~(e-ie). For, let w

The trigonometric series for cp(6) conjugate to ~(e) is (reference 12)

cp(e)=~(-^ - ansinne+bncos nQ) n=

Setting bll+ ian = Cn, we have m

——..... ... .. .... .....--- —- -----.. ——. - —..—-.--—- --------- -.-..-..—-———

——.—--— ...... .. ...., .. , —-, ,,., ..--—., ,!..- —.-.,........... ,..—................. ,,.. -. -.,. .. .—.

18 NACA TM No. 1156

z [

=f(~)=-~ (a-b)t + (m-b) ~ !! 1

so that

k= (^) - ~ (a+b); kl = - ~ (a-b) 2

By formulas (5a) and (6a), section 4, we obtain immediately

Further, we have

[

a2-bz

1

--if(c) ~(;)=-; i (a2-b2) ~2 + 2(a2+b2) + ‘—

[2 —

By (8) section 4, we obtain

If b = O the ellipse degenerates into a plate of width 2a and then

As a second example we shall consider the problem of deter- mining the potential flow of an infinite fluid for any motion within it of a Joukowsky wing, (^) Incidentally, we shall compare certain geometric characteristics of Joukowsky profiles which we require for determining the hydrodynamic forces.

As is known (reference 13) the outer region of the circle K, (fig. 4) in the plane Z1 = xl + iyl, is conformallymapped on the external region of the Joukowsky wing in the plan z!. x’+iyl with the aid of the function

NACA TM No. 1156 (^19)

The JoukowsQ wing is given completely by the parameters a, a, R, the geometric meaning of which on the z~-planeis clear from figure 4. The angle u may be taken as a number characterizing the concavityof the profile; c = R cos u-l characterizes the thiclmess of the profile; if c = O we have an arc of a circle. For R. const.,^ a = const.,^ and the variable a in the Z1 and

z: planes vary similarly, hence the number a determines only the linear scale of the Joukowsky profile. The point Ml transforms into the sharp edge of the wing M. (^) At this point the correspond- ence will be quasi-conformal. (^) All anglea in the zt-plane will be twice as large as the corresponding angles in the dently at the point

zl-plane. Evl- M1 the direction MIP1 w1ll correspond to the direction M P and the direction of the radius OMIN1 goes over into the direction of the principal tangent at the inflection point M of the Joukowsky profile. Hence ~PMN = 2ZP1M1N1 = Za.

Transforming the irrotational flow about the cylinder K with the aid of relation (l), we obtain the flow about the Joukowsky profile, the angle between the velocity at infinity and the real axis in both planes being the same since (dz’/dzl)m = 1/2. The point ‘ will be a critical branch point of the flow if the angle of inclination of the velocity at infinity to the xl-axis is equal to a or fi+a. Only in this case will the velocity of the transformed flow at point M be finite and therefore only in this case is an irrotational motion of a potential flow about a Joukowsky profile possible. Evidently the direction of a possible irrota- tional flow makes the angle u with the direction of the principal tangent at the sharp edge. This direction is known as the first axis of the profile. The angle u may thus be^ implicitly defined with respect to the profile as the angle between the priacipal tangent and the first axis indicating by definition th~ Li:ection of a possible potential irrotational flow about a Joukmsky pro- file. In what follows we shall assume the direction of tinefirst axis as the real axis of a Cartesian system of coordinates (^) XY) the origin of which is at the sharp edge and we shall set x+ty = z. Evidently

z =^ zle^ -^ ia (2)