Unsteady motion of a flat plate in a cavity flow

Office of Naval Research Department of the Navy Contract No,

_N62558-2269

Unsteady motion of a flat plate in a cavity _flow

by

JA. Geurst

Reproduction in whole or in part is permitted for _{any purpose} of the United States Government

June 1 959.

Instjtuut voor Toegepaste Wiskuride Technische Hogeschool te

Delft Nederland.

C 0NTEI'TS.

Introduction

Part I. Reentrant-jet flow with _{unsteady perturbations0} Summary

The steady flow

Li..

The unsteady motion as a perturbation of the steady flow Additional conditions

First order approximation for _{small breadth}

Part II. Linearized theory of the unsteady cavity flow Summary

Linearized theory of the steady flow

The unsteady flow; method of _{the complex velocity} Linearized unsteady cavity flow _{as a first order} approximation

11 _{The unsteady flow; method of the complex acceleration} potential

The drag

2.

1. Introduction.

The publ±catjons about the unsteady motion of a body in _{a fluid} with free streamlines _{can be divided roughly into three categories.}

The first category deals _{with exact solutions. In that case it} is necessary to restrict the _{field to instationary motions of a}

special ty-pe. Voii Krmàn [i.] _{was the first, who}

_took

_{that direction,}

Later on publications appeared by _{Gilbarg [2] and Woods [3].} Under the second category we will class those articles, in which the unsteady motion is _{considered as a perturbation of the}

steady flow. Woods [Li.] _{investigated in that way the instationary}

motion of a flat plate according _{to the Helrnholtz,-Kjrchhoff}

model. Wu gave in [5] a formulation of a boundary-value prOblem _{for the} Roshko model (also named _{transition model). Closely}

connected with this is the research on the stability of free streamline _flows. Ablo.w and Hayes [6] published _{a report on the subject,Thejr}

results were extended by Fox and Morgan in [7].

The third category contains _{papers, in which the whole flow is} linearized. Tulin [8] _{was the first, who applied the}

linearization on a steady flow, _{arkin [9] solved the unsteady problem of an} in±inite cavity behind a flat plate under incidence and _{gave a} formulation of the finite cavity _{case. Wu developed in [-lOjand [5].} a theory f or theuneteady finite _{cavity case, Tiirnan [1-1] and}

the

author [12] treated the _{case of a partially cavitated flat}

plate

under incidence. The unsteady motion is considered small with respect to the steady flow (second _{linearization). In order to}

get a closed cavity they assumed a discontinuity in the unsteady _vertical pertur-bation velocity in the wake _{of the body. Another possibility is to}

abandon the closure of the cavity for the unsteady _{case and to} re-quire that the normal.component of the unsteady perturbation velocity

3.

should be Continuous in the wake. As a result of a discussion with Dr. Eisenberg and Dr. _{Tulin this last possibility}

will be investi-gated in this report by comparing the linearized cavity flow _with the corresponding reentrant-jet flow. For a steady cavity flow it has been proved, that _{linearized theorrjs}

a first order approxi-mation to reentrant -jet _{flow theory [13] . From the following it} will appear that also in _{the unsteady case linearized theory can be} formulated in such a way that it becomes a first-order _{approximation} to reentrant-jet theory. _{We will prove this connection, just as in}

[13]

, for the siiiplest

case of a flat plate, moving perpethicular to its plane. The _{research splits up into two}

parts.

In the first part _{we consider'the reentrant-jet}

flow with un-steady disturbances9 As _{a result of the linearization}

we can assume this motion to be _{harmonic in time without}

loss of generality.

In the second part we treat the linearized problem. _{The solution} is obtained by two different methods. The _{first method uses the}

complex velocity and is _{directly connected with}

the first part of our research. The second _{method starts with the}

complex accelera±jon potential an

offers the best perspectives for more complicated problems9

Finally it should be

remarked that we have omitted in this introduction papers, _{that deal with the}

beginning of the _unsteady motion. Several Russian authors have worked in _{that direction e.g.}

Gurevich [iLi.]. Their _{investigations are directed towards the}

deter-mination of the cavity _{induced mass (see}

Birkhoff and _Zarantdnello [15]). In this report however, we are _{interested in the}

instationary motion as a whole.

I. Reentrant-jet flow with unsteady perturbations. 2. Summar

A flat plate is placed in a flow of an incompressible non-viscous fluid with uniform velocity perpendicular to the plate. Behind _the plate a cavity with reentrant

jet has formed. The jet is supposed to disappear on a second sheet of _{the physical plane considered as a} Riernann surface. This artifice is used to represent within the framework _{of potential theory the dissipation of energy, that is} observed at the rear end and _{in the interior of the}

cavity, (see Bjrkhoff and Zarantonello {15]). _{We prefer here th}

reentrant-jet model above the Riabouchinsky.model, _{since the former shows}

more resemblance to experimental _{observations (see Eisenberg [i6]).When} the plate performs an unsteady motion, th _{shape and size of the} cavity will change. If _{we assume that the deviation from the}

steady position. is small in _{a certain sense, this unsteady motion can be} considered as a perturbat.jon of _{the original steady flow} (lineari-zation of the unsteady effects),

We treat first the stationary flow _{Next we investigate the effect} of the unsteady motion.

3e The steady flow.

The steady symmetric flow _{about a perpendicular flat} reentrant jet is shown in fig. I

Bv

fig. I.

plate with

5.

The Cartesian coordinate-system has been chosen in such a way that the middle of the plate lies in the _{origin and that the velocity of} the fluid at infinity has the direction _{of the positive x-axis. We} denote half the breadth of the plate by b. We assume the fluid to be incompressible and non-viscous and the motion to be steady and

irrotational. The relation between the _{pressure and velocity is} then given by the Bernoullian law:

p+pU2

=

p±pq2

+

(3.1)

where q magnitude of velocity

p pressure

U velocity at infinity p pressure at infinity

velocity on the cavity

PC constant pressure on the cavity

p constant density of the fluid

We choose the units of length and time _{in such a way, that the}

constant fluid velocity q0 _{on the cavity is equal to I The} suscep-tibility of the fluid for the _{occurrence of cavitation is characterized.} by the cavitation number cr, that is _{defined by}

p -p

o 00

(3.2)

1

Prom (3.1) and (3.2) follows that

equations

= 0

ox

oy

+ - 0

ox

0y continuity equation irrotationality

(3.Li)

These equations express, that w = u-iv is a holomorphic function _of z=x-i-iy in the interIor of the fluid. On _{account of (3.Lj.) we can define} a velocity potential _{c and a stream function ? by}

u = (i+oj _(3.3)

Let the velocity vector _{= (u,v), where u and v are the x- and} y-cornponents of the fluid velocity. u and. V have to satisfy the

(a) 0

(c,a)

(do)

fig.

2.

We now determine, in a) _{and b) (see below)}

respectively and w as a function of t. The _{conformal mapping i.e.}

z as a function of t is then given by 6.

U=-

y

ôy

= -

(3.5) 'Then

= c+ip

is a holomorphic function of z=x+iy, and

w=a

_(3,6)

At the moment we consider only the _{upper half of the physical}

plane i.e. the upper half _{of the region taken by the fluid0 As}

a result of the presence of the reentrant jet _{this region is}

two-sheeted. Since the region is

_{simp1y-connec,}

we can map it conforrnally onto the upper-half of a t-plane (t= _cT+iT).,

Remark: this o is as a matter of

course different from the a- defined

b

_(3.2)...

This will cause no confusion in the following. The images of _{3 points on the boundary}

can be fixed arbitrarily. We choose the image of _{Z=oo as t = oo}

the image of A as t = -a and the image of B as t = 0. a is a given positive

number (see fig. 2).

clz=w'1 d =w'

dt dt

or z

=fw

K dt _t-c where K is a real constant

and _{= K [ t-(d-c) log (t-c) ] + constant}

7.

(3.8)

(3.9)

b) Determination _{of w as a function of t.} We introducefl _{log w = log q-ii9' , where}

is the angle between the velocity vector and the positive x-axis. fL is _{a holomorphic function} of t in the upper half-plane. has to satisfy the _{following mixed} boundary conditions:

o°<o<a

=Q

or

ImS=0

-a

<o-< 0

_or

Irn!l=-0 < o < c _{q = I or ReS1 = 0} (3.10) c

<cr<d

=-7

_or

_Im2=jr

d

<o<oo

9=Q

_or

_irnLO

For the solution of this boundary value problem we continue IL analy-tically into the lower half-plane _{by means of the reflection}

principle of Schwarz £2.(t) (3.11) a) Determination of _{as a function of t.} We consider .

is a holomorphic function of t in the upper half-plane. Since the real _{axis is the image of streamlines (where}

= constant), _{is real along the real axis, By means of the}

re-flectjon principle of _{Schwarz we can continue}

analytically into the lower half-plane,

can have singularities on the real _axis. These can be _{no essential singularities (for}

a proof see Birkhoff and Zarantonello [15] _{). Singularities can only occur at t= and. t=c.} From a consideration of the local behaviour fllows, _{that remains} finite at t=oo _{and behaves like}

at .t=ó. As a result of the

branching of the streamlines _{at the point D,}

will have a siniple

8.

The boundary value problem is _{now equivalent with thefollowing}

Riemam-i-Hi]jjer _{problem (see Muskhelishvili [i 7]).}

Determine .cL(t), holomorphjc in the _{upper and lower half-plane, with}

the following linear rel.tions.on the real axis.

oo<cT<

a - a

< T

< 0 - 7rj +

O<O<c

1+=O

(3.12) c < 0 < d. - _{= 2 gj} +

d<o-<00

ü

and STL _{denote the boundary values .l assumes in approaching the}

real axis resp. from the _{upper and lower side.}

Moreover Si has to be continuous _{at the pOints t = 0 and t = c. At} infinity Si must remain finite.

We consider first the homogeneous _{Riemarin-Hilbert problem i.e. the} above mentioned problem with the _{right hand sides in (3.12) replaced}

by 0. A solution _{of this homogeneous problem is.}

= ft(t-c)

(3.13)

The cut in order to make the _{scjuare root single-valued is taken} along the real axis from t = 0 to t = c. The branch is chosen in such a way that

trt(tc) t

as too

The Riemann-Eilbert problem (3.12) _{can now be formulated as follows} Determine Si(t) in such _{a way that' on the real axis th'e following}

relations are fulfilled

oo<cr<--a

_{{-].-i-] =0}

- a < o < 0

+7j

< 0- < d

After some calculations we arrive at

fl=- log

w=

-a t

.11k

_27i

[]

- [ci

₌

_{__}

+ log C ['fd( t-c) ft(d-c)] and Iid(t-c)-Tt(4-c)1 [Tt(c+a)fa(t-)] ' [Td(t-c)ft(d-c)]

The expression for z as a function of t can now be written _as

z =K

f4G

d$.

td-c

dt

J _{[It(c+a)_fa(t_c)]2 [tfd(t_c)4t(dc)]} t-c

9.

(3018 )

Remark: (3.16) and

_(3.17)

represent _{and w also in the lower half} of the physical plane when we map _{it. coaformally onto the lower} half of the t-plane by _{means of (3.18).}

In the solution,which is the combination _of

_(3.17)

_and

_(3.18)

_,three unknown constants viz. K, c and d. still occur. These unknowns are

determined by the following conditions

i ) at t _{oo we must have w = U = (i -for}

z has to be a single-valued function of t

the distance between A and B = b = half the breadth of the flat plate.

We first formulate conditions 2) and 3) _{mathematically..}

The condition that z should be _{a single-valued. function of t, reads} e _{dt = 0}

(3.19)

d < cr<oo

₌₀

(3.1 L)

By theans of the formulas of Plemelj (see _{Mukhe1ishvilj [i71)we find}

0 _d. fL=a/t(t_c) _t.[

_f

+

f

2i d

_(3.15)

27V1j

_o-o)(o-t)

J {'ft(c+a) -fa( t- )]

Condition

₃₎

_{can be written as}

fedt=i

-a

We now apply the _{.3 conditiOns to the solution we have found.} Condition i gives us

(i +o) rc+a-'f]

[J3.

Condition 2) furnishes the _equation

Cf

_+c=0

Prom condition

₃₎

_{we obtain}

0 .1 K

I [L±.ac-c

2 dT-c-Th- d-c = bj _(3.2L.)

I

_{[!o(ca).4a(T_c)]2} {d(-c)--T(d-c)]

OC

Elimination of between

_(3.22)

and

_(3.23)

_{gies the following} relation between c and

o.

(3.25)

1+0- =

[j1

+11+

Moreover it can be derived _that

ry

[i-

/

-1

_(3.26)

and

d-c =

(3.27)

The integral in _{(3.2)-i.) can be expressed in elementary}

functions,Sjnce we don't need ouch an.expre5sion in connection with the purpose of _our research and since the expression _{would become rather unwieldy,}

we

omit the expression here.

10.

where C is a closed contour _{surrOunding the plate and cavity i.e.} surrounding that part of the real _{axis that lies between -a and d.} Application of the theorem of residues on the exterior of C giveS

res. e

dt

too

= 0

(3.20)

(3.21)

(3.22)

It.

(3.25)

gives the relation between c and o-.. Prom

_(3.27)

_{follows the} connection between d en o. Finally (3.2L1.)determines K as a. fuii.ction

of b and a-. Thus the solution for the steady case. is completely

determined.

The unstead motion as a êrturbation of the steady flow. jwT.

Let the flat plate perform an oscillation.of the form _{he normal} to the ma±ri flaw. ByT we denote the time variable, h is the ampli-tude of the oscillation. The dev±ation is taken positive in the

direction of the negative x-axis, Thus the normal velocity _{in the}

jw7

direction of the negative x-axis is equal to jwh e

Remark: it is asEumed tacitly that of all expressions, in which _occur exponential terms complex with respect to j, we must take the real part in order to get the true time behaviour0 Since _{we linearize the} unsteady part of the motion, this procedure will cause no troubles. Finally as a matter of sourse the imaginary units i and j are

dis-connected; we have i2 -I and = -I , but ij is not reducible..

We now suppose that Wh is small with respect to unity. _{In that case} the unsteady motion can be considered as a perturbation of the _steady flow. We will neglect, as usual, second and higher _{powers in the} dis-turbance against first order terms.

As a result of the unsteady motion c and d will become _functions of the time T, Also the boundary conditions will change

a) On the flat plate, i.e. _{on the interval -a < a-< 0 of the real axis} of the t-plane, we will now have in a first order approximation

=

e'

_(Li..i)

(see fig. 3), where q0 is the magnitude of velocity in the unperturbed steady flow.

1L

SWy

fig. _3.

Remark: quantities referring to _{the steady floW will be denoted by} the suffix 0, and these referring _{to the first order terms of the} unsteady perturbation by the suffix _1.

b) On the boundary _{of the cavity, i.e. at 0}

_{< 0}

_{< C,}

the instationary Bernoullian law will now be true:

+

q2 + = c(T) _(L.2)

We put

=

o + and p = p0 + p1.

In a first order approximation (L1.,2) _{can be written as}

+ q + q02( -)+ + C(T)

or as ±

q02

- + - = D(T).

q1

If we put

= r , _{the Bernoullian law becomes}

0 1

a OT

p

± q r1 ±

= DT)

_(L1.5)

Remark: At the partial _{differentiation against T the space variables} are kept constant; this amounts to the fact, _{that rp0 and are kept} constant.

On the boundary of the cavity we have p1 _{= 0, thus}

+ qr1 = D(T)

oh the cavity _Li..6)

12.

We apply this equation on the boundary of the unperturbed _{cavity. The} corresponding points are found by orthogonal _{projection. In that case}

(L.6) is in a first _{order approximation replaced by}

+ r1 = D(T) _{on the unperturbed cavity}

13.

since .there %=1. Moreover _{on the boundary of the unperturbed} cavity it is true that

= constant

On differentiation of (L1..7) against we get 0.

We will show first that

lj

_oo

-r and

I

2) r1 _{- i&1 is a holomorphic function}

of t, with = Putting = + have d. dz _W (

o)

-

-

dz - - I _.1

In a first order approximatio

w = e = e a I

= e°(1cL1) =

_w0(I+1)

or

L.1

=-w0 _1°

Always in the _{seine order of approximation}

= log w = log q - i& =

log q0(i

- i& -= log + r1

-- i1

+ r

-Thus = r1

- is a holornorphic function of t.

Prom (L.IO), (L.i2) and (1.iL) it follows finally that

= ₌

r

q.e.d. _(.15)

On the unperturbed cavity the boundary _{condition now reads}

as

(Lj..iG)

or r1 =

(Lj..i 7)

i.e. r1 _{is a function of T-cp}

In our case of harmonic _{lime dependence we have}

j w (T-cø)

r1 = g e

where g is a still

_unknown

constant, real with respect to 1, but possibly complex with _{respect to j.}

a

vnw'

F

-(L1.,9)

ILl..

We anticipate here this harmOnic time dependence. The justification follows only by (5.1.5), since it _{expresses g linearly in h.}

For S1 ₌ Sl + _{the following mixed boundary value problem can now}

be formulated.

Determine fl.(t), holomorphic in the _{upper half pland, in such a way}

that it assumes on the real axis the _{following boundary values}

<o-<-

a

_orlmfl=o

- a<

o-<

0

oo<cr<a

-

a < cr< 0

0 < 0

<

c

C < O- < d

d<0<oo

+

e'

_{or Im} =

-

jwh eT

2 q0 ₂ ci.

jw(T-)

_{j(T-p ).}

0

< o

< c _{log q0+r1 =g e or Re l = g e} 0 (Li..i 9)

c<cr<d

=-7r

orImS=+7r

d<0<oo

₌₀

_orlmfl=O

Putting

fL(t)

_{= T3T}

_{according to the Schwarzian reflection}

principle we can transform the mixed boundary value _{problem into a}

Riemann-Hubert problem with the following _{linear relations along the real axIs}

-a<o-<O

_{_fL...ti_2j.2eJ0}

-

.

w(T..p)

0<

cr<

c Sl _+fl

= 2ge

+

c<o-<d

fi -Si

=27ri.

d<0<oo

Remark: g can be taken equal _{to zero at c < 0- <}

c if c0 < c. Letting c=c0+c1 and d=d0+d1, _{this Riemann-Hilbert}

problem (Ll..20) can

be split up into the following

Riemann-Hjlbert problems for 11p and

çL where fl =flp

PLp-

cLp _{= 0} -ALp

7r1

+ Sip

_{+fLp =0}

clp _{- SaD = 22ti} - flp _{= 0 and} (Ll..20) (Lj..21 )

00<0<-a

ci+SL_O

_S S

-a<o-<Q

s S q

o <

o- _{<, c0} fL+ ₊ =

2ge'

C

<0<oo

fl+fLO

0 _{S S}

According to (3.16), the solution _of

_(Lj.2i)

_is

= ..

L(c±aHcJJ.

_{+ log} L[d(t-à)ftLd-c)i

(Li..23)

{Tt(c+a)fa(t-c)] _{[fd(t-c)t(d-c)]} We now determine the solution _{of (Lt..22). By means of the}

solution of the homogeneous problem _{= t(t-c0) the R.H.-problem (L1.22)}

can be written as follows - <

0- < -

a

-a<< 0

0<o<

₀

-a

ci

-=0

cih+

[fd(c-.r)

_+'I-o-(d-c)]

Jo(c0-o) (o-t)

(L1.22) (L..2L1.)

(L..25)

(L.26)

Using the formulas of Plemelj (see [16]) we get finally ci=

_{et(t-c )}

J _{[a(c-0-)-[-0-(c+a)]}

[Id0(C-o)-J-d-(d-c)]

c _,jWK[cr-(d

-C )

_{log (c} .o-)]

____ -

e AJt(t_c e 0

I-cr(c0-cr)(o-t)

C0 < 0- <

00 It

atO<o-<cwehave

= .K [-.(c10-c0) log (c0-)] and by virtue of (3.17) at - a < < 0

[Ja(c0)(ca)]

'I = -2i jaa 2ge

1.6. From _{(Li..23) and (L...27) we}

find

=

t

a

d1

ft(t-c0)

_/

d

+ .i_wh

et(t-c)

Ed(:_)-(d-c)]

J

_{[IdC-)J(d-c)]}

'dcr

_-

_{£ ei°7'rit(tcjf} -(d0-c0) log

(c0-c)]

(Li..28)

5

Additional conditjohs.

In the expression (L..28) for _{three unknowns still occur viz. g,}

C1 _{and d1 ; g is a constant, possibly complex with respect to j, since}

we have already anticipated the time behaviour _{of GT-cD); the two} other unknowns, C1 and d1 _{are functions of T, complex with respect to} j, which will appear to be of _{the exponential type too. All}

three

quantities are real with respect to 1. For their determination _we have the following three _{additional conditions:}

i) The perturbation velocity _{must vanish at infinity i.e.}

= .0 as Iti

+o

_(5.1)

This gives us as a result of the symmetry of the problem only _one condition real with respect to i.

2) The perturbation cannot have _{a source or sink at infinity. This} condition is equivalent to the requirement, that the _{pressure should} remain finite at infinity Moreover _{it is the generalization of}

the

closure condition to the unsteady _{case...Mathematically it is} formu-lated by

res. S?1 _{= 0}

as

It!

(5.2)

The same remark as under i) _applies..

.3) Finally we have to _{ensure that the presaure on th}

cavity is not only constant in space, but _{also constant in time. This}

last condition is not expressed by the boundary condition, since the boundary

17.

condition (Li..i.6) has been obtained _{by partial differentiation against}

We shall now formulate _{condition 3) mathematically.}

The instationary Bernoullian _{law reads according to (Li..5) as} 2

+ q.0 r1

-+ -'= D(T)

we have D(T) 0.

Prom -- = r _{follows that on tl2e real axis}

LçD I d. = r1 d =

f

r dcr = 00

-t-d.

=Ir K- °dLr=Re

I

K°dt

I cr-c0 J I t-c1 (.5.3) We define _{=0 as o-}

_forT

_{0; cr and}

are the coordinates in the t-plane. We _{are allowed to do tha,t, ince is determined} except for a constant and since _{is finitea-L infinity as a result} of the absence of a source there. Since st infinity r1 _{= 0 and p1=0,}

(5.Li.)

where the path of integration in the last integial can be deformed according to Cauchy's theorem,

On the cavity the following identities _{are valid.}

jw(T-ç)

r1 =ge

_{-, p1 =0.}

Thus (5..3) can be written on the cavity as

jw(T-ç)

or using (5.L1) as

t-d jw(T- ₎

ReKf...00dt=_ge

00

This is the mathematical form of _{the third additional condition.}

It

is easy to prove that this condition _{is independent of the choice of} t in the above mentioned interval.

We have seen now that the 3 additional conditions are linear in the unknowns, Since the known members _{of the equations have harmonic} time dependence, we can put

c1 =c1 e

_{,d1 =d e}

If we

Re

Kf

For

GT-cp0)

_{we have anticipated this time beha;iour,} remarked. c and _{are real with respect to}

complex with respect to j.

gives us

c

with Q(o-) =

Applying (5.2) and using (5.8) _{we arrive at}

ti

--- +

c0+a

['!a(c).f(c+a)]

hfQ)

a

'fo-(c0o-) d

Al

_d

___ -

d1f

_ç;:;

From (5.8) and (5.9) it can be derived that

fd

d A +

_-Q ](2

_-1) 2

_c+a

_d-ó

c 0 0 0 0 =

P 1(°-)

(d0-oi

d l-cr(c0-r)

d0dJ d.

0 d -c 0 0 0 [rfd0 ( c0-cr) [Td0(c-a-) -jwK{cr -(d0-c0 (.2 _{-i) =} 0 0 0 -a

10-jWK{0(d-c) log (c0-)]

_(-)

(c0-)

+tfd-c)]

1 8. as already i, but in general (.7) (5,9) log (c0o)] crdo 1cr(c0-cr) -jwK[0-(d -c )log(c0-o)j (d0-o)dø-0 0 (5.11) =0 o t <

We shall now use the _{3 additional conditions for the}

determination of the 3 unknovm constants, c', d _{and. g. Application of (5.1)}

let L1. = e ,

(5.6)

goes over into

t-d t-c dt = w.

ge

-K[ t-(d0--c0) log (c0-t)] 7r e

=0

(5.8)

Using (5.10) and (51iYwe find =

_t(tco)f

Q -jWK[0-(d-c) log (c-o-)] o-(c-o-)(o-t) (t-c) (t-d) - _{t(t_co)f°}

The first integral in (5.12) _{can be expressed in. elementary functions.} We will not do that, since in view of the purpose of our research _we

don't need such an expression. We can now apply condition 3) _as expressed by 'ormu1a (5.7), We take first _{t=o in (5.7) and se} formula (5.12) to substitute for Sl, _{The left hand member of}

.7)

now replaced by a repeated integral. Since _{the integrand of this} integral is except for the endpoint t = 0 a holomorphic function of t, the integration with respect to t _{can be performed along a path,} which meets the interval [-a, c0] of _{the real axis only in the point}

t = 0. We now move the _{endpoint of the path of integration from t=0} to a neighbouring point t

= C,

_{with Im e 0. Since both integrals} in the repeated integral _{are now absolutely convergent, we}

can change the order of integration. Evaluating the inner integral and letting afterwards

C -

0, which is legitimate _{on account of the uniform} convergence

K[-2f

-[-+

-jwK[cr-(cl-c)

of the outer integral, we get

d -c

do--jwK[cr-(d0-c0)log I, Jt(t-c)dt Re urn I e-+OJ _(t-c)2(o--t) 0

I.Q(

d

-C0 -a (c-o-)] (ci0-o-) log log (c -o-)] 0 rt 27 L -arcs1n (-0

Use has been made of the following _identities

C 2 [[c_o-_iT _] [w

c-7+Af;J

do- + 19. (5.1 2) =

e0

log C0 (5.13)

[[c -o--'T&]

_o-log 0 0

[Wc4

at

7

< 0

and

2 . 2o- / 0- ₁

= - + i. _{-arcsin (s- - )]} ' _{at 0 < o- < .c}

0 0 0 0

The last integral in (5.13) can be reduced as follows

fC0

-K[cr-(d -c _{) log (c0-o-)]} .

e ° °

_{[-arsn (- -1)}

_{do- =}

0

=

f[

_{-arcsin ( -1)] d}

e_T_

Q00) log (c0-r)]

0. = -jwK(d0-c0)log C0 c -iwK[o--(d0-c0)log(c0-o-)]

WKe

+f-. J 'io'(c0-o-) 6, First order a finally as

roximation for small breadth.

S.

20.

(5.13a)

(.5.1 L)

0

The integral occurring in the last member of (5.ILi.) and the integral of the same type in (5.13) can be expressed in confluent

hyper-geometric functions0

Using (5.ILi.) we can write (5.13)

9 d -o- ₉ d -o- [i c

-jwh [ 2 Q(o-) - -- do- + f Q(o-) - log

0

do-]

J tJ-O(c j 00

0-[Wc0-r-"f]

0

Formula (5.15) gives us g; c and d1 are then found from (5.10) and (5.11). The solution is now completely _determined.

c0

wK[(d)1(]

---_

.[2(d0)-

] = 0.

+ g

f

e

a) We determine first the first _{order approximation for the steady} flow. For that purpoe we put K = I and consider first a as unknown instead of K. Formula (3.2Li.) gives then by means of (3.25) and (3.26) or (3.27) the relation between a on the one hand and b and _{0- on the} other hand. We shall now consider a and.c _{as given and determine b} and o- as functions of a and c. We _{suppose a to be small, It will} appear that then for fixed c _{P alsob is small, i.e. as a tends to} zero, also b tends tQ zero. The cavitation humber o- as a function of a and c is given by (3.25).We expand this expression into _{powers of}

La

-

1

-

+

ad-c

i±

]2

Lj.ac(c+a)(i-y2)

_(6'3)

I C d c+a _1+y

Ec(iy)2

ay2

where

Ja(c,)-

(ca

_{is the new variable of integration.}

sfà(c-o-)+ 'i-cr(c+a)

According to (3.26) and (3.27) we have in a first order approximation

and

d-

C

=0

Hence in a second order approximation the _{expression in (6.3) becomes}

I

equal to La

f

y2

ly

dy

=

(7rL)

_(6.5)

.(1+y)

In the

same

order of approximation we find for b

b =

L)

_(6.6)

Expanding expression (3.17) for

_{w into ascending powers of,we get}

w=i

±...

(6.7) and using (6.2) a [tJa(c...iy)....

w=1

-0- c+a c+a)] 0- r t-c

T

o'-d

[Jd(c-)-

tf.o(d_c)] 0--c

+[:L.]

t-c

In a first order approximation it is also _true

thatdz=dt

Hence in that approximation w is given by

w=1

+

...

_(6.8)

21

1 +o=

_+Li:F+...

_(6i)

Hence we have in a first order approximation

(62)

We now expand the integral in

_{(3.2Li.) into}

_{powers of , using} (3.26) of (3.27). For that _{purpose we introduce first a new}

variable of integration and

take

the real part of the integral.

A I (5.10) yields c' F

jh

_(i-+2)

F

Co

-functions C 0 0

f

e&r

J0 'o-(c-.c- -C

= jw

e -r

_/°

t t-c0 tc t I -b-c0

t-c0

j

Co U 0 CO - jw 2 J(w -=) I

C -0

0 o--t ±a

-_j

0 C0

t(t-c)(t-c)

jW0-,) _.pii dO

Relation (5.15) between h and g gives in the same approximation

2(c

0)

--2jWh(7+2)

'T

c.. + g

f

-2-__W

y dy (i )2 e 1) C 0 YCC

-;p)----'

_C e 2 _[JJct.)+jj

(w.f)J

2 L) 2 22.

The relation between b,, c and 0 reads

= 2b

(6.11)

Expansion (6.iO) is uniformly valid _{in the interIor of the physical}

plane.

j3) First order _{approximation for the unsteady motion.}

We first expand _{into ascending powers of}

- . Introducing

again y as a new variable of integration _{by means of (6.3a), we find.} from (5.12) in a first ordei _{approximation}

(6.-i 2)

= 0

_(6.-13)

(6.iL.)

The integrals in (6.ii) and (6.-12) _{caiibe expressed in cylindrical}

23.

Hence (6.13) can be written as

-2jwh _{C0 +} ge 2 [J0(w )(c0 -

+ j c0

1 (w _)]=o

(6.17)

and

(6.lLi.) as

= 2h

_c - gc e

[J(w

)+j

(W)(618)

Combination of

_(6.17)

_{and (6.i8) gives}

'C

= g e 2 J0(w

_(6.19)

where use has been made of (6,6) in _{order to eliminate a.}

For d1 _{we find the same expression as for}

In the foregoing we have supposed _{that < 1, but also that} Wh

-

; In other words

-

cannot be taken to small, _since

Co

otherwise we should have taken into account higher order approximations for the consideration of the unsteady effects i.e. we should have

calculated the coefficients of higher powers of 'T in the expansions,

II..Linearjzed theory of the unsteady _{cavity flow.}

u1ary

In the linearisd theory the flat _{plate is replaced by a}

singularity for the complex velocity. _{It will appoar, that for the} steady flow the expression for the _{comolex veloc:.ty is identical} with the first order term in (6.iO). _{It i pcssile then by means} of the calculations in

6a.)

_{to express the strength of-e}

singularity in half the breadth b of the flat Dlate _{This will be done in 8).}

Assuming that the unsteady effects _{can he considered as small} perturbations of the linearized steady _{flow (second Unearizatioii),}

a

we can proceed in a maniier analogous to that fcr the steady óase It

wl± again

_{appear that the unsteady part of the flow can be idenifi;:.} with the first order term of fL1, _{s derived in 6 ). In this cose too} the unsteady part of the strength or _{the ringurity can be brought} into connection with the amplitude _{h and the circular frequency w of} the harmonic oscillation of the flat plate. The calculations _{can be} found in 9) and iO).

Finally we show that the application _{of the acceleration} poten-tial methodmon the unsteady cavity _{flow gives the same results}

as the foregoing treatment by means of the complex 7eloctv.

8. Linearized theor Of the steady f1ow

We suppose that the flow pattern _{as a result of the presence of the} flat plate can be considered as a small _{erthrbtion of the original} uniform flow field0 This will generlly be the case if the breadth 2b of the plate is small with respect to the cavity length0 The plate can then be replaced by a singularity. The _{rear end of the cavity} will show a similar singularity. Within _{the framework of linearized}

theory the occurrence of these singularities can be inferred from the presence of a stagnation point in the neighbourhood. We shall not go into further details but refer to [18].

We choose the Cartesian coordinate system in such a way that the cavity extends between the points (o,o) and (.,o). The velocity at infinity has the direction of the positive x-axis The units of length and time are chosen such, that the constant velocity

along the cavity is equal to I We put = (i +u,v). Since 1-U < I , we have u < I and v < I We will neglect second and higher powers of

u and v. (linearization)0 w = u-iv is a holomorphic function of

z=x+iy.

The linearized Bernoullian law reads

PPc=PU

(8.1)

Hence we derive u = - (8.2"

00 2

On the cavity we have u = 0, _(8.3)

Since the maximum width of the cvit.y is o± the same order of magni-tude as the perturbation velocities, we can apply condition

(83)

on the x-axis between (0,0) and. (e,o) instead of on the cavity. The holomorphic function w has now to satisfy the following mixed boundary value problem (see fig. Li.).

= 0

25.

( oJ

(t0

fig. L.

Determine v(z), .olomorphic in the upper half of the z-plane, such that it assumes the following boundary values on the real axis:

-00<

x< 0

v=O

_orlmw=0

O<x<e

u=O

orRew=0

(8Lj)

26,

By means of the Schwarzian reflection principle this boundary value problem for w(z) can be transformed _{into a Riemam-i--Hilbert problem} with the following linear _{relations along the real axis.}

0< x<

< x< 0

< x <

The general solution with intëgrable _{singularities at z=0 and z=e}

is

(8.6)

The cut for both square roots _{is made from 0 to £; the.branches}

are

chosen such that

asz-c

_(8.7)

We assume that the strength of the singularity at z=0, determined by the flat plate, is given. We have _there

wA1I

_(8.8)

Hence the strength of the _{singularity = AJ7}

This gives us one relation between A and For the determination of the tmknowns A, B and _{e we stll need two relations, These}

are

furnished by the following _{additional conditions.}

-i ₎ w = - at z= according to (8.2)

(8.9)

2)

res. w= 0 at z =c

_(8io)

This condition is equivalent to the _{closure of the cavity (see [18]).} From (8.9) we deduce

and from

(8.10)

A - B= 0

_(8.12)

These equations result n

+ w - w = 0 +

w +w =0

+

w -w =0

(8.5)

Thus w have for w the formula

This expression agrees with the first order term in

(6.i 0

identify c and _{From (6.9) follows that}

27.

(8.iL.)

if we

Afe 2b _{(8.1 5)}

This is the connection between the strength _{of the singularity at}

z = d

and the breadth 2b of the plate. We have _{shown now that}

linearized theory can be understood as a first order approximation of reentrant-jet theory. The same statement holds as to the

Riabouchinsky-flow. (see [13]).

,9..The unstead flow method of the corn lex velocit

The coordinate system and the units of length and time are chosen as under 8). As a result of the harmonic oscillation of _{the flat} plate (amplitude h, circular frequency w), the _{strength of the} singulaiity at z = 0 will vary harmonicly with _{time. We assume that} the amplitude of the time dependent part _{is small with respect to the} time independent part, but large with respect to second and higher powers of the last one. This assumption _{enables us to treat the}

unsteady effects as linear perturbations of the linearized steady flow (second linearization). Hence all quantities can be split up addi-tively in their steady and unsteady parts. We _{put accordingly}

A u = u0 + u1 with u1 = U1 e3 A _jwt

v=v+v1

_{v1=v1 e}

A = A0 + A1 A1 = _A7 jWt _(9.1)

B = B0

+

B1 = B

jwt e = + = e

et

A jwt e

Remark: the time variable is denoted _{as usually by t instead of by T,} as we did above As a result of the unsteady motion _{the boundary}

28.

condition on the cavity will changes For _{its derivation we start} with the linearized Euler _{equations of motion f or an ideal fluid:}

+_u

ôt ôx

..x

-at

ôx

_1E

p ox

On the cavity we have p _PC

According to (9.2) we get there Ou Ou + = Substituting (9.1 ) we arrive at ,' 0u1

j

w 1 + = 0

The general solution of this _{differ-ntial equation is} ul

thus =

0.

the following mixed boundary _{value problem.} in the upper half of the _{z-plane and}

00 <

< 0

O<x<e

<x<0o

The solution is w

= -

e jwt

fz

(z )

I

1'dx

(9,2)

real with respect to 1, but possibly

+

w -w =0

+

-

A. w + w = 2geU w - w = 0

x(ex)(xz)

(9.7)

+B

[.

₍₉

8)

assuming the following boundary _{values on the real axis}

00<

x<O

_v=O

_orlmw0

0 < x. < e

u=gei°:t)

_{or Re w} =

(9.6)

e

v= 0

or Imw=O

By means of the reflection _{principle of Schwarz this}

boundary. value

problem can be transfol'med into _{a Riemami-IIjThert problem with the} following linear relations along _{the real axis:}

A

where g is an arbitrary constant, complex with respect to _je

For w = w0 + w1 we now have Determine w(z), holomorphic

29.

We now divide this expression for w into its steady part w0 and its unsteady part w1. We find

w0=A0J-2+B/

z-e ₀

(9.9)

w=_eWtiz(z_e )r

_-A

_+B _1

I

ooJ

ix(ex)(xz)

o 2 0 2

z-e0 z-e0

I

z-+A'

2

I z

For the relation between A0 and the breadth _{2b of the flat plate,} the determination of B0 and the _{connection between and the} cavitation number 0 we refer to 8).

Assuming that in (9i0) the strength of _{the singularity at z=0 is} given, we still need 3 other conditions for the determination of

/\ A

the Li. unknowns, g, e

-and B1 _{These conditions read as follows}

i) as under 8) we must have

0

w=-

&t=oo

or WI = 0

This condition means that the Derturbatjon _{velocities vanish at} infinity,

res. w = 0

at z

or according to 8)

res. w1 0 at

z =

00 _(9.12)

This condition means that there is no source or sink at infinity. We know already that there is no vortex at infinity as a result of the

smnyietry of the problem. As a result of this condition the pressure remains finite at infinity in the unstesdy _{case toO..}

Boundary condition (.9.5) on the cavity _{has been obtained by}

partial differentiation against x _{Heice it is possible that a time} dependent additive function with a constant as a special case has been differentiated away. In any case it is necessary to connect the

-'-au Z oo

pJ _(9.10)

30.

pressure on the cavity with the pressure at infinity. _{This is the} third condition0 We formulate it now mathematically.

According to (9.2) we have

= -

p

op1 ouI

= -

p

(j w

UI

The unsteady motion is not allowed to disturb the connection between the pressure at infinity and. the pressure on the cavity, i.e. we have to require that

P1,

-P1,=

0 _(9.11-i.) or by virtue of (9.13), that +L) 0

f(

w. + )dx = 0 (9.15) 00 Integrating we obtain +0

j

wf

u1dx

= -

i'

(+0,0) _-

_(9.16)

or

/

i dx = (9.17) 00

Application of (9.11) and (9.12) gives respectively

A D and

+Aj

+

=0

dx +

©e

x dx] :

Wx(e-x)

A _I' A A

Since A0 = B0 - , it follows that

e0=O

(9.1 8)

(9.1 9)

By means of (9,10) and (9.20) we find

A

A

J(e

I

-x(e0-x) / . 8

We can now apply (9.17)

f(e0-)def

-

mf

(-x)(-x)

(9.22)

The inversion of the order of integration can be justified in the same way as in 5). We now use the following ideitities

0 0

I

[arcsin0+))f

dx 2

J

'-x(e0-x)(-x)

i(e0)

2

(-x)(e0-x)

£0

(9.23)

Integrating partially we get from (9.22)

e

.,\ o-W

_/e3Xd

0J

J.x(e-x)

31.. (9.21)

By means of (6.i 5) the integral in this formula can be expressed in a Besse function,

This yields

3W A JW

Lj.

=ge

Remark: it is possible to write the solution (9.8) of the boundary value problem (9.6) at once in the following form

w=et

i_J

z(z-e)

. x-z

By putting A=A0 and B=B0, the conditions (9.11) and (9.12) _are automatically satisfied. Finally it. is possible to get from (9.22)

to (9.2L1.) withdut evaluating the first integral in (9.23). Intht case one uses the fact, that the integral can be considered as the

solution of a certain boundary value problem.

(9.26)

A

o-eJW

Lj. (9.2L1.)

?r(e

(i 0.2) We have already identified, in 8) the following quantities _{viz. t and}

z;

c0

and

e0.

_{If we now can identify also g and g}

, the two

expres-sions (10.1) and (10.2) become _{the same. Since however,}

we have

assumed that the strength of the _{singularity for WI at z = 0 is given} and is equal to the strength of the cinularity for _{at t=0, we} infer from both formulas that _{indeed, g and g must be}

identified. This gives the proof of the statement, that for the unsteady _case too linearized cavity theory is a first order approximation _of reentrant-jet theory.

From (6.19) and (925) We see that also C1 and must be identified,

Finally we derive _{an expression for the strength of the}

singula-rity at z = 0 in terms of h and W. _{This expression can most easily be}

derived from (6.12). According _{to that formula has the following} behaviour near t = 0

- jwh (-t _(10.3)

0'

U

32.

-1 0. Linearized unstead cavjt flow as a first order_approximation.

1'

We shall derive expressions for _{and w1} , which can be compared in

order to show that WI is a first order approximation to S.

For that purpose we express SL1 _{in terms of g, and WI in terms of g.}

From (6.12) and (6.13) we find for

9'o_.

/'(c-r)do - I I e

_______ +

I t(t-c )

₀₀

I

C -joxJ-+

Q)fl'

. F

t(tc

)P

"Icr(c,,cr)

Combination of (9.10), (9.20) and (9.2L1) gives us

A e _A

_j

g

I _P e ' + I _r e LI D I

-rz(z_e

') 2) 7r 1 / (i 0.1 )

function by means of p-pc

-

-Hence (11 .1 ) can be written as

+ =.

a-b ôL. ox

(11

.3)

at ax Oy

is called the acceleration potential. As _{a result of the} incom-pressfbility and irrotationality cp is a harmonic function. We can

now introduce its conjugate harmonic function _{according to} ax - ay

8y Ox

Now F(z) = + _{is a holomorphic function of z=x+iy,}

for which we have ]? ow aw az - at + (11 .2) (11 .5)

F(z) is the complx _{acceleration potential. We may formulate the}

linearized cavity problem of 9) _{now as the following mixed boundary} value problem for F(z).

3.

where (-t is positive for real t _{< 0. Application of (6.6) gives}

us

'.-.

_)h2J

-t)

(i o.Li.)

11 The iinstead. _{flow method of the com.lex. acceleration otential.}

The coordinate system and. the units of length and time are chosen as under 9). We start from the linearized _{Euler equations}

ôt ôx ,ô;x

(11 .1)

ôt ôx

pay

In these equations we have linearized _{with respect to =i , the} velocity on the cavity in the steady _{case. We introduce the}

where a and b are quantities,

_{only dependent on t and real with}

respect to 1. Remark: this

_{a is different from the a used in}

_{part I.}

According to the second

_{linearization we can write}

F = F

+ F1

_{with F1}

_{= F1}

et

A

jwt

a=a+a1

a1 =a1 e

_(11.8)

b = b0 + b1

_b1

_{= b'}

e0)t

We identify F0 and

_{w0. This is in accordance with the}

_foregoing

equations. Thus

= -

_H'

.

+PJ.j]

(11.9)

For F1 we now get the expression

2. F

= -a

- -=.---

_{+ b}

₊ 1 °

2tIz(ze)

02

Jz(z-e0)

z-+ a1

_{+ b}

_where

_=e

-

(11.16)

In this expression for F1

_{3 unknowns viz, 7, a1}

_and

_{still occur,}

For their determination

_{we have the following 3 conditions}

i)

F1

= 0

at z =

(11 .11)

i.e. the perturbation pressure vanishes at infinity.

2) the velocity must be a single-valued functions

As a matter of

_{course this requiremeritis fulfilled}

_{for the}

horizontal component of the

_{perturbation velocity u1 by virtue of}

the symmetry of the problem:

_{there is no vortex sheet in}

_{the wake.}

F(z) = a 'J

_z

+ b

z-e

(11

.7)

Zi

Determine F(z), holomorphic in the

_{upper half of the zpine}

and assuming on the real axis the

_{fpllowing boundary values}

oo <x<0

_orlmp=0

0

< x< e

_{cp =0}

orReP=0

_{(ii .6)}

e

< x<oo

=0

orlmF=O

C

71

-'.5.

So condition 2) _{expresses the fact, that there is no line of} discontinuity for the vertical.component _{of the perturbation}

velocity v1

3)

_{the relation connecting the unsteady part of the singularity}

for P(z) at z _{= 0 and the given unsteady part of the singularity for} w(z) there.

As to

₃₎

_{we will prose that the unsteady parts of the}

singularities for F(z) and w(z) at z = 0 are the same.

According to (11 .5) ve have

A +

o

-

'I

from which follows that Z

-

e I - J oc 00 = (z)

-f

e

_F(C)dC

_(11.13)

Use has been made of (ii .11 )

-Since the integral in the last member _{of (11.13) is finite at z=0,} A

WI and F1 have the same behaviour at z=0. q.e0d.

Equating the strngths of the two singularities we obtain

A a2. A

a1 + _{= A1J + A0}

(11 .iL)

This is the mathematical expression of condition

_3).

Application of condition i) gives us

(-ii

.is)

We shall now formulate condition 2) _{mathematically.}

Using

_(11.13)

_{we can assert that the single-valuedness of w}

is

equivalent with

e F1(c)dC - Q

(ii.i6)

where C is a contour surrounding the _{cut from 0 to e in the z-plane.} For (z) is already _{a single-valued fction.}

Since a = A0 = - _{, we can write}

F(Z) =a

-+i_

A _A o2. _a -

J

...j..

-.8

Z

where use has been _{made of (11 .1 5).}

Coiidjtjon (11 .1 6) _{now reads as}

A j jwz oX jw a1 e

0 j

dz +

- fe

(-)

'+ Cf

z-eo

+ j JI

kfz(zC)

z Lj. dz z-.e

e0x

x dz = 0 A or jWCp

A______

a1 0

r

dx ± _J eJWX X

e-x

-

Wx(-x)

Replacing in the _{integrals x by e0-x as variable} Co

"

f

dx

_°i

jW

a1e0 ₊

_J

- tfx(C0x)

or using (6.15) and (6.i6)

A o-7

jw

(w2)

+-_j

[

From (9.20) _{and (9.2L) we can derive}

A e _e A1

J(w

2) +

[J(w

_Q)

+ j

j1 (w

2)]

dx dx = 0 C 0 2

=0

0

Combination

of (11.iL), (11.21)

_{and (11.22)}

yields

A A A

a1 =A1 andA1

_=e1

This means that both methods, viz, _{the method of the} acceleration potential _{and the method of the complex}

lead to. the _{same results. ThIs is}

important, because (11 .19) of intwgratjon _{we get} (11.20) (11 022) (11.23) complex velocity, the method of the complex acceleration potential _{is preferable in}

more complicated problems.

iTrg0

We calculate _{the drag by applring the}

mornentui _{theorem on the interior}

of the contour indicated in fig, 5.

.36.

C

fig.

_5.

.c represents the linearized version of the flat plate. The _whole contour is denoted by C. We use the method of the _{complex velocity.}

It is clear by virtue of the symmetry with _{respect to the x-axis} that the component of the force perpendicular to _{the main stream}

vanishes. Sinee the horizontal sections of C give no contribution to the x-component of the force and the contribution _{of the vertical} end sections can be made arbitrarily small by making _{the contour} thinner, the momentum theorem can be written as

(p-p)d.y = (1+u)v dx

_{(i 2.1 )}

C

where p is the pressure exerted by the fluid on the flat plate and the direction of C and C is chosen positive i.e. such that the interior of C lies on the left.

Since the drag D _{= P (p-p)dy, we have}

at z=O

(i 2.2)

w_i[ljwhaejwt]z_

_atz=

_{(i 2.3)}

C

370

D=p

(1+u)vc=-Imp

wdz-Imp

ImP /Tfrdz=_rpRe

res. w21

z=O

By virtue of (8.15)

wo - I (I and by virtue of (IO.)-i)

WI

-±jWh---i

v+L 2 A _{7C-F2 2b} at z=O Hence we have

Substituting in (12.2) we find

D = p 2b Yr [1+2jwh

eJt

]

and

CD = [i+2jwh

e]

₍₁₂₅₎

--pU 2b

In (12.5) we have neglected the factOr (i-f) in _{accordance with the} order of approximation we have tied throughout. In [13] we have proved that for the steady case the insertion of the factor (1

)

- already known as the (i ) rule - gives the correct next order of

approximation. 13. Conclusions.

In the foregoing sections we have shown that linearized cavity theory can be formulated in suh a way that it becomes a first order approximation to reent±ant-jet theory. For the proof we have chosen the simple exomplC of a flat plate normal to the main stream _direction since in that case the calculations are _{as simple as possible. This} choice does .ot restrict the validity of our results. For it may be safely assumed that the behaviour at the end of _{the cavity, which is} the important point in the steady _{as well as in the unsteady case,} will not be influenced by the shape of the body in front. After

linearization the flat platebocmeasingalarjty, _{as one would expect.} We have assthned that the unsteady motion is small with respect o the steady part of the .distubance field caued by the body and the cavity. For the non-linear theory i.e. the _{reentrant-jet theory,} this assumption was necessary to attack the problem in _{a reasonable} way, for the linearized theory it enabled us to produce explicit results.. Moreover we may expect that the condition of small unsteady

I

We have shown that for the linearized

_{cavity flow the method}

of the complex acceleration potential leads to the same result as

the method of the complex velocity,

_{provided appropriate additional}

conditions are applied. In the

case of the second method these

conditions can be derived easily

_{as a result of the linearization}

by comparing the linearized flow

_{with the reentrant-jet flow. A}

condition, which seems to be

_{new, is the one relating the pressure}

at infinity and the pressure on the cavity. From our analysis it is

clear that in the case of the former

_{method, i.e. the method of the}

complex acceleration potential,

_{it must berequ±red explicitly that}

the normal component of the

_{perturbation velocity should be continuous}

in the wake. Since this condition

_{has been overlooked hitherto,}

_we

lay stress upon it here,

In addition the above linearization of

_{the flat plate into a}

singularity, which may be considered

_{as a slight generalization of}

linearized theory as used up till

now, enables us to treat more

com-plicated problems in a relatively simple

_{way, E.g. the effects of}

gravity on a cavity has been dealt with recently in that way.

In following reports

_{we will treat the unsteady cavity flo* past}

a lifting profile, f or the partially

_{as well as for the fully}

cavitated case, by the methods outlined

_{in this report.}

Acknowledgement.

The author is indebted to Prof. dr. R.

_{Timnmnan for interesting}

discussions and for reading and criticizing

_{the manuscript.}

L.Os

References.

[i] Th. von Kármàn, Accelerated flbw of an incompressible fluid with

wake formation, Annali d.i Mat0

_{pura ed appi. IV, 2.(19L..9),}

2Li7 - 2L.9,

D. Gilbarg, Unsteady flows with fre boundaries,

ZOA.M,P,,

_(i952),

3L1 _Li.2,

L.C, Woods, Unsteady cavitating flow

_{past curved obstacles;}

A.R.C. Techn0 Report C.P. No. 1L1.9

_(i95Li)0

[Lj.]

L.C. Woods, Unsteady plane flow

_{past curved obstacles with}

infinite wakes, Proc0 Roy. Soc. London,

A

_{(i 955), 1 52 - 1 80.}

T.Y. Wu, Unsteady super cavitating

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_1958.

C,M. Ablow and W.D. Hayes, Perturbation

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Techn. Report, Graduate Division f

_{or Appl.Math.,Brown}

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(195L.), Li.39-L,56.

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1953.

B,R, Parkin, Fully cavitating

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California Institute of Techn. Report

_{no. 85-2,Pasadena, 1 957.}

T.Y. Wu, A linearized theory for

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California Inst. of Techn. Report

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[11 3

R. Tiniman, A general linearized

theory for cavitating hydrofthjls

in nonsteady flow, Second Symposium

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_a

partially cavitated profile (in Dutch),

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_{Hogeschool te Deift,}

Rapport TW I

,

1958.

.

J.A. Geurst, Der Zusamenhang zwischen den linearen und

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_{Kavitation, GAIVIM-Tagung}

Hannover, 1 959.

[iLi-] LI. Gurevich, The impact on a lemma with discontinuous

a

U

41% [15] G. Birkhoff and E.H. Zarantonello, Jets, wakes and cavities,

New York 1 _957.

P. Eisenberg, On the _{mechanism and prevention of}

cavitation, David W. Taylor Model Basin _{Report 71 2,}

1 950.

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