An extension of woods theory for simulating the change of immersion of a cavitating hydrofoil

SHIP HYDRODYNAMICS 'LABORATOR:.'

OTANtEMI.

'FINLAND,

--REPORT NO 19

., AN ...EXTO,ISION OF' yooDs THEORY FOR SIMULATING' THE

. .

° CHANGE: OF IMMERSION .,OF A CAVITATING HYDROFOIL

ISSNO356-!1:313

iSBft9517514.68-0

surface or a hydrofoil in a.seaway change their _immersion as a function of time Consequently the cavitation number is alsdtime7dependent: This type of flow closely:-resembles that of an unsteady ventilated cavity. _{In the} present paper theories developed to investigate the ifistability.of ventilated cavities and unsteady plane flows with infinite wakes are applied to the

caseof a

fully cavitatingplatewith.a,changing _{caVitating'number.} The boundary conditions are formulated, and the corresponding .solution is Obtained. _{in .principle, the} SolUtion.thus-obtained Could form the basis for estimating the 'probability . of thecollapse of the cavity.

No numerical. results, nor comparisons with eperitents are available yet.

ABSTRACT CONTENTS, NOTATION

1. INTRODUCTION

2.. THE BOUNDARY VALUE :PROBLEM

4.: THE SOLUTION To THE BOUNDARY STEADY CAVITATING FLOW, liNSTEADY'CAVITATING FLOW :bRAG AND LIFT:

7. CONCLUSION ACKNOWLEDGEMENTS -REFERENCES 'APPENDIX A -,APPENDIX- B APPENDIX & ,

Integrals for evaluating Integrals for evaluating

Integrals,Jor

evaluating

LIST OF 'FIGURES

--Fig.. 1: Evolution of a three-stage cavity:: Fig 2 Dependence of Uc at sectionC,17C2.

the cavity pressure with time

Fig. 3. The physical Model

Fig. The independent variables 'Fig. 5.. Convection of cavity volume

11 17 26 31 38 148 50 51 52 514 57 63

NOTATION

circulation per unit angle

drag coefficient

lift coefficient

length of section

ca distance of the stagnation point from half chord

E' mathematical abbreviation defined by Eq. (3.1)

mathematical abbreviation defined by

Eq. (3.1)

mathematical abbreviation generally

fsubscript

Gc

=ccO' defined by Eq.

(2.8)

Gw = aw,-E2wO' defined by Eq. (2.16)

I . . abbreviation for an integral

subscript

unit imaginary number referred to space

unit imaginary number referred to time

real constant

empirical constant

two-sided Laplace transform

P. length of cavity

N1 defined by Eq. (3.7)

N2; defined by Eq. (3.8)

pressure

variable in the Laplace-transform plane

constant corresponding to the sink at infinity

Q the rate at which the unsteady component of

cavity volume is passing into the wake

velocity magnitude

oyc

distance

measured,

along a streamline: time

advance velocity

of

:section

the Unit function

. .

_

. . integration variable'in Ch. 4 cavity Voilime: integration variable in Ch 4 i, cOMpldk'streaMfunation cdOrdinate cdOrdinate

Width Of teddy Wake

x. +

-geometric angle of:Attack.

defined by E.g.

V.YsI

Ys2?

circulation

_coordinate

of

.transformed plane jump in 0 across the wake'

jiimp'in

= {a

.10

unsteady increment to the diSpleCement.

. -coordinate of tranSforted plane

0. across the wake

1

Oh

Tysir iterationi'Variable in Ch:

subscriptparameter defining the decay of the wake-amplitude

4

0 angle between q and Ox-axis

angular measure of the unsteady wake displacement

1

2 YS1 YS2)

integer

real frequency in radians per second

variable defined by Eq. (2.11)

fluid density

defined by Eq.. .(3.2)

.cavitation number

natural cavitation number before cavitation

'Csr variable, defined by Eq. (2.12)

time scales defined by Eq. (2,12)

(1) velocity potential'

X 0 + i0, dependent complex variable

0 stream function

iteration variable in Ch, 4

. 2 = ln(U/q), speed variable

wc frequency

SUBSCRIPTS

refers to a point on the boundary

refers to a point on the cavity

.refers to the imaginary part

refers to pressure changes

sur refers to a point on the pressure side of the foil

refers to a point in the wake.

integer

refers to the variable part of a quantity

1 refers to the suction side of the wake or cavity

refers to the pressure side of the wake or cavity

mean steady value

refers to a point at infinity

capital letters refer to positions given in the figures

OTHERS

{_} = l I

(

complex amplitude {a a

eiarcawe)1

we

it _YS2

f

-

f

one writes f(E) = 0(6(E)) as E 0, ,if lim [f(E)/6(E)1 <

E÷0

1. INTRODUCTION

The unsteady features of a cavity flow can be described by the concept of re-entrant jet /8, pp. 161...169, 185/1. This re-entrant cavity model has been used by Woods /17/ when developing a theory for the resonating frequencies observed in the experiments of Silberman & 3ong /12/. The following elaboration of the physics of the flow is given by Woods /17, pp. 437...438/2:

"The cavity that forms behind a submerged body moving at speed through a liquid is continuously supplied with gas and vapour from the cavity walls, a mixture that leaves by etrainment in the turbulent foam and liquid usually found at the end of cavity and introduced into it by a re-entrant jet. Bubbles of gas and vapour pass

down-stream in the wake and are quickly reabsorbed into the liquid. Despite some fluctuation in the re-entrant, under steady flow conditions this natural cavity is quite stable, having a constant shape and a constant cavity pressure _{pc, approximately equal to the vapour} pressure of the liquid. The most important parameter descriptive of the flow is the cavitation number

1

a =

(p.-pc)/(7 pU2), where p. and U are the pres-sures and velocity in the undisturbed flow well upstream of the body and p is the liquid density. Now, while

a

can be controlled through p. and U, in some experimental circumstances it is of greater convenience to alter pc through the introduction of air or other

gas into the cavity through the base of the body. _This

Numbers in brackets designate References at the end

of the paper.

In this paper the References are used directly as

widely as possible. _{In this case the corresponding} part will be put within quotation marks. _Different notations and the special case under consideration make some changes and adjustments necessary.

1

12

reduces a, and a so-called ventilated cavity will result when the air supply rate Q is large enough to reduce a to the cavitation point.

The relationship between Q, a and av - the natu-ral cavitation number before cavitation - has been

examined in an extensive set of experiments by Silberman & Song. However, for a critical value of a/av below

0.2 the ventilated cavity starts to vibrate violently, changing its length and width periodically, and the cavity pressure pc

oscillates about p, which

denotes the mean value. The re-entrant jet is com-pletely suppressed during that part of the cycle in which the cavity length is a maximum. Further increase in Q has no immediate effect on a/av; the oscillations become more violent, but their frequency remains

con-stant. However, when a still higher value of Q is

achieved, the cavity quickly inceases in average length and pressure, and a/av suddenly drops. The new cavity vibrates less vigorously than the first, and its surface now carries a wave-form of two wave-lengths that is convected downstream with the local stream velocity

qc = UT/TTU (U is assumed to be constant).

The key to the phenomenon just described lies in the way that vibrating cavities part with their cavity air. Steady entrainment no longer occurs, but is replaced by a fission in which the last wave is periodically pinched off at a time when the cavity pressure is too low to sustain a cavity of greater than average length."

The flow around a cavitating propeller blade section behind a ship hull has some similar features. For example the cavity number changes as a function of the angular position of the blade section. Thus a two-dimensional foil with a changing cavitation number could simulate the time dependent variable immersion of a propeller blade section.

This simulating model will be investigated in the present

paper. The essential assumption now is, that the cavities

part with their cavity air by the same mechanism as de-scribed above. In this initial survey the emphasis will be on the formulation of the problem, and on appraising the mathematical feasibility of the solution. To simplify the mathematics only the flat plate section is considered. The approach can be extended to other unsteady cavity problemS One straightforward application would be a hydrofoil in a seaway.

Figures 1 and 2 depict four stages in a complete cycle for the three-stage cavity. This three-stage model is adopted from Ref. /17/. The main differences are, that according to experimental observations, the slope of the cavity surface on the pressure side, u2, is always zero. Also the time scale in Figs. 1 and 2 is defined differently

{by Eq. (2.12)}. The mean velocity over the cavity, cic0' is, as usual, given by

1 2

r

=

qcO/U

=

Ir-770'

(50 =

(P,o-Pc0)/(7 PU )'

After pointing out these differences, full use can again be made of Ref. /17. pp. 439...440/:

"The process shown in Figs. 1 and 2 is an idealization

based on the description given by Silberman & Song. At

u1 = 0, when the last wave is just pinched off, the re-entrant jet commences strongly, shooting water into the cavity at such a rate that the air supplied by the body together with this water is more than sufficient to keep pace with the expansion of the cavity caused by the convection downstream of the next wave. The result is that the cavity pressure increases steadily from the low value it achieved on the previous cycle to a maximum

at T = 2/6, at which stage the re-entrant jet is considerably weakened and the air-water foam is being

14

removed by entrainment just as in steady flow. Now the continuing lengthening of the cavity occurs with very little new water being added to it, so that the pressure

falls. _{This reduced pressure decrees a narrower and}

shorter cavity - a consequence of the usual steady-state theory - and this is achieved by a repetition of the pinching-off process. The cycle then repeats.

Also shown in Fig. 2 is the dependence on T of uC' the slope of the cavity suction-side surface at the section C1-C2.3 The logging of

(1c-pc0) behind u by 900 is an important experimental result, and supports the only phenomenological element we shall introduce into the theory. _{One minor point is that, as shown in} the figure, we have chosen as the reference length X not the mean cavity length but the shortest length in the cycle. The reason is that Silberman & Song found experimentally that the minimum length is closest to the length found for non-vibrating cavities at the _same value of a."

In the present case the cavity pressure is a time-de-pendent function given by

jv t

(1.2)

PC _1. c0 + pcwe 2

where v is the real frequency in radians _{per second.} This frequency is the same as the first harmonic of _the wake variation. _{The corresponding cavitation number is}

(1.3) a = iv t -Pc0 -pcwe P T---T-pU2 7 pU

7

ivnt

-

a, + a e U 3

When used as subscripts capital letters refer to positions given in the figures.

where a is the complex amplitude, and p is the fluid density. In this way the effect of the angular position of the blade section is simulated. Now the "air supply rate" is not known beforehand, but it is to be determined as part of the solution. A non-zero air supply rate implies that the cavity is not closed. The mathematical reasoning of this point of view is as follows /10, p. 134/:

"In problems involving a free surface, cavity-closure conditions are usually not employed, undoubtedly because of the difficulty of applying them."

Problems dealing with foils beneath a free surface are usually treated by applying Tulin's double spiral vortex termination concept /15, p. 21/, /14/, /11/. The double spiral vortex model and Woods' approach have some common features /10, p. 131/:

"The free streamlines bounding the cavity end in a pair of double-spiral vortices. ... There the flow speed discontinuously jumps from the cavity-free-streamline speed qc to the undisturbed speed. U."

This jump of velocity is also included in Fig. 3 of Ref. /17, p. 441/. Further the solution space in the paper of Woods /17/ is a simply connected region /15, p. 21/. The present approach differs from Tulin's double spiral vortex in one respect. _{The boundary conditions are defined} in such a way that no net force is left _{over to act on} the wake /4, p. 39/, /5, p. 345/.

The obtain a solution some unknown parameters must be included in the equations. One of them is the "air supply rate" just mentioned. _{All these parameters are fully}

determined by the boundary conditions. Then the solution gives the length of the cavity, and the pressure fluctua-tions on the foil. A complete evaluation of the present

neW y of 'attacking the problem. is only posib1e by

,

2. THE BOUNDARY VALUE PROBLEM

The physical model of the flow is depicted in Fig. 3. The free, surfaces

S1C1 and S2C2 are at a constant pres-sure at a given instant of time, with subscripts 1 and 2 referring to the suction and pressure sides of the foil or wake respectively. In unsteady flow they are composed of material lines rather than streamlines /18, p. 125/.

At this stage the following restrictions must be imposed on the model and the

solution:-The free sUrfaces are assumed to be stream surfaces. The solution is meaningful, i.e., excepting the region of spiral vortices, it is only singly covered. The

streamlines, cannot cross each other /15, p. 24/.

ay (defined by Eq. (2.3)) is assumed to be small. In Chapter 5 it will prove nedessary to neglect second order unsteady components.

The present problem will be treated by extending the results of Woods /17/.../21/. In the following they will be drawn on as widely as possible. Ref. /7/ is another example of such an extension.

"Let

a

denote the flow direction relative to the Ox-axis, the direction of the flow at infinity.; then in the steady flow the value of 0 on the streamline joining the rear of the cavity C1 to k = co, say ul,

will be zero; but in the unsteady flow 01 need not be zero (see Fig. 3). The function 01(5), whei,e s is distance measured along a streamline, is introduced to account for the unsteady component of the wake- displace-ment thickness due to the convection of bubbles down-stream of

C1' While the underlying steady stream of

bubbles will displace streamlines outwards, it will not contribute to

u1" One restriction must e imposed on 1' namely that the total volume of the cavity, and the

18

perturbation volume in the wake, must, at most, increase at a constant rate. This means that the source at infinity will be time-independent, as required by the theorem analogous to Helmholtzis.theorem on conservation of circulation. Without this restriction the pressure at inifinity is unbounded. _{Hence the perturbation in} the sum of the wake and cavity volumes is zero. _Now the waves, or other perturbations in the wake, may be assumed to be convected downstream with velocity U

without any change of shape, apart from a possible decay in amplitude. Thus

(2.1) ul(s,t) = exp[ _ci)/U),

-cu1(s-sC1)1111(sC1,t-(s-s

u2(s,t) = 0 (or eu2 >> 1),

C1 being the value of s at the cavity end C1 /17, p. 441/."

Later on it will be seen that the resulting problem can be formulated in a simple way, only if it is assumed that

Eu2

= m.

Hence the shorter notation

eu1 =u is adopted.

"In addition to the restriction on 1)1 imposed by the

time independence of the sink at infinity, a singularity at

C1 is required in a model for the resonance shown in Fig. 1 /17, p. 441/. _{While C1 is a fixed point,}

the cavity length is free to change by the introduction of ul. If there are no bubbles in the wake, ul will be localized near s _{= sC1' and in this case its effect} can be incorporated in a singularity at Cl. The point

C1 is the point chosen to mark the change from one type of boundary condition to the other /17, p. 444/.

(2.2i) 2 = ln(U/q), i.e. q =

and

(2.2ii)

x = a + io =

ln(U[dz/dw1), i.e. Udz = eNdw,

where w

= 0 + up

is the complex stream function and z = x + iy. In Eq. (2.2) w, not z, is chosen as the independent variable, so, in accordance with the

linearization mentioned above, w is the stream function of the mean steady flow /17, p. 441/."

Due to the asymmetry of the flow (and the existence of lift) the mixed boundary value problem must be defined and solved in coordinates used by Woods in Ref. /18, p. 126/. The frame and the independent variables are shown in Fig. 4.

"With the approximations at the beginning of this chapter, the curves AS1C1 and AS2C2 can be taken to lie on the streamline iP= 0 and so arrive at the w-plane. Because of circulation the points C1 and C2

do not correspond in the w-plane. The t..-plane is mapped

into the semi-infinite

strip

-Tr < r, 0

< n <

= y + in)

by

(2.3) w = 2a(cos a -cos(C+a )+(C+2a )sin a),

where w = 0, y = -2a _{defines the position of the} front stagnation point A. The clockwise circulation is /4, p. 39/

(2.4)

Oci - 0

C2 = ima sin ay.

The surfaces of the foil and cavity are mapped on to

n = 0, while the trailing edge streamlire _CiD1. is mapped on to the curve nsin a + sinh n sin(y+a ) = 0.

CurveS lie close to y pr., so thaOhebOuildarY conditions on C1D1:, can be.:applied..on y-=

Little terror /1.8, p, 126/,"

:7./*

assumption :of a, constant cavity length' isconsidere.0_{. .} as-a-justification-for-taking

S1 j.,:.and,-,(§2,

as

tine-inde-,

-pendent:y This iS-_-thephysical reasoning,.thar the SO=Called.

--.1"mappAngapprOximation" of Ref. /21.,-p.17/ cah.be 4plied in the present case The mapping approximatiOn7is

, _

-can be IT..eplacedHby , where thesubscriW {q} J-cefers-. to the-mcan-stcadY' value. The mean value of the independenr variable w onAthe cavity surface is

:Later on, when changing Variables] this formula

useful.

,

The presSur-e is. given by the Bernoulli equation. . i.e.

2 22 +.1 2 1

(2,6)-:' P L

OT

1)0-

pcio ,= p4:3

(2.6) thc'integ-ration has

_{beenperformed-along:.:2'4;-="}

,

-from = /17,,.p. A42/. On.

the

cavity surface one has

(a+1)

Denoting

=- dO qcods = rUds.

c2-q ' -2cCW

and with* instantaneous constant presSure within the

--, '., -.- ' ,...,. .,,.

caviti-;the differenLiation of the linearized Bernoulli

.. - ., eAUatiOn

lea:ds

7.6/.19, p, 1-1.591.' : , .., (2.8ii)7-- {q2,G } + {G } = D a. do ou c 3t c 0

and on integration

1 1

(2.9) G_c1(0_0't) = G_c1(0_S1,0't 2 0 + 2 0S1,0)

qc0 4c0

on the free surface,

Gc1(00't) = 0 otherwise.

A similar expression is valid for Gc2. Changing variables (Eq. (2.5)) Eq. (2.1) takes the equivalent form

1 (2.10) u1(00,t) = exp[-e0(.0-(PC1 ,0) 11)1 (4)C1 ,O't 7(410-4'C1,0)) on the wake, u1(0 0't) = 0 otherwise, where

1/esubscript is the damping length.

The independent variables

(2.11)

will also be needed in Chapter 5. _{Defining "reduced" time} scales

, 2

'cic0 (2.12i) T

1 °CIO -4)S1,0

and the variables

, 2 'cic0

t, T,

t, 4.C2 ,0 CPS2,0 (2.1211) qi = T1/r2, .72 = 2/r2, 31 = a2 = 72-E2+1, Eq. (2.9) becomes (2.13) G01(C1,T1) = Gc1(-1

,Ti-C

-1) on the cavity. E = 1

(0

1,0

-1(0

2 C1,0 1 +0_S1,0))/(1(0 2 C1,0 1 -0 S1,0)), E2 =

(0

_2,0 --40_{2 C2,0}+0_S2,0))/( 2 C2,0-0S2,0)) ,

:Eq. .(2.10) can be expressed in the same,wa

0.110=

.1- 1! A'

...1

,

on

the_wake.." The difference.

_{14C.1,07.S1,01}

(2°15), - 4)S1 0 a0{cos a' 4:dos. SiMilar. formulae-., G- Q - 0 for

v,=

1i2,. wv .1.1v w0v . and (2.16i1) l'w0v.{U2Gwv} :lt(Gwv)-The-,,..subscript (2.14)

yields

In :the.ma(hematical model described

- ,

:1-41-'1he example under investigat..icri:(defined'bY.Eqs,:.

..(1'.2).and(1l.'3)4111o1-4ons of long duration are treated...c

A redaced frcquency. .

.1.8)"

'

sin y san

ayo+

(777YS1),c/y0Y: exist

-also in the

wake,

-iaobtained.from:EA.

refers to the wake. Comparison.with Eq. ,

characterizes the flow.

After these preliminaries the boundary conditions for x as an analytic function of are written down sepa-rately for the steady flow and the superimposed unsteady

part. The subscript {b} refers to the boundary and {w}

to the unsteady part. The two boundary conditions are

(2.20)

nbO(Y) = ln(1+a0), in < y <1S2' YS1 < Y n = 0,

0b0 + 7TU(y+2ay0), in Y.S2 < Y < YS1' n = 02

where

U(y+2ay0) is the unit function,

lim x(Yin) = 0, -to 0(+7+in) = 0, 0 < n < c°, e(-n+in) = 0, 0 < n < 02, = 0, 0 < n

< w,

sgTr+in) 0, 0 < n < n+in

f dz = i(the width of the steady wake} = iywo,

11+:in r+indw df c141 = "C1,0c/IC2,01 -n+in and (2.21) 1

10

ejwcT1 bw a (i) = , in < Y < YS2' YS1 < Y < 2 1+a 0 where a = -cwp /{-2 pU2 }, n = 0,

lim'O'.:(+Tr+in)..='.0'

r14.-(-7r+in) =:

w

(+n+in) - 0 .(-ii l.in)

_7r+in:._-dz = 0, 0 24 0, 0, n <

-sink at.

infinity:-(.66

Nyd

respectively /18,

P.

126/._

In'thO,abOVe; Q represents, the

a known function of y.

is 4 known funCtion of y. The Cbw is a Carrec-ion term -due'tO,iWmavement of the stagnation point which is

initially unknown /21, p, 163/. Condition (iii) follows

--frOm-,the:chOsen.values, of q,, U

and 0;=0

The.

-m4theMaticalmodel assumes that no timedependent.-.Sinki exists it infinny (iv):.

_{Kelvin's CirculOtionjhe4ean=':'}

"

car-not be applied directly t6 the present problem _{as the tdke} 'prelUdes 'contours enclosiqg the obstacle. For the-iun,

In(ii) - 1:).0

steady increment the following consideration is. valid. The flow into the wake is replaced by a distribution of steady sources and vortices such that the shape of the free stream-lines for the basic steady flow is preserved (v) /21, p.

159/. The slope of the cavity surface was discussed when

writing down in Eq. (2.1)(vi). In (vii) the unsteadiness causes vorticity of fluctuating strength to _{be shed from} points el and C2. The mathematical model assumes a finite wake thickness (viii). In (ix) a difference in 0

at the end of cavity ensures that noi Zet force is left over to act on the wake /4, p. 39/.

"The point r has to move with the fluid if the

right-hand side is to be independent of time. _{If (ix) is to} be satisfied it is necessary to assume that

-i(v+a ) i(44a ) dw 1 (2.22) - iae Y -271(4)C1-4)C2)-10

iae 5

5 + 0(Qe12i9

at infinity, i.e. near C

= im.

When Q = 0, Eq. (2.22) is in agreement with Eq, (2.3) and if it is _{assumed that} the relation between the _{z- and C-planes is exactly as} indicated in Fig. 4, the approximation of replacing material lines by free streamlines is equivalent _to neglecting the effect of Q on the form of Eq. (2.3) takes on n = 0, or on /18, pp. 127...128/

(2.23) 0 = 2afcos a -cos(y+a )+(y+2a )sin /

In general 0 will depend rather weakly on Q, and the theory to be given below neglects this dependence."

26

"T,HM :SOLUTION TO THE BOUNDARY VALUE PROBLEM

.1.1* .solution to the boundary value 'problem :defined

by

the boundary ..Conditions Si ) ,:

(ii)

,

(vi), and (vii) is. obtained by extending the equations derived by Woods , /19,

pp. 126;' 131, 148.. :1.:5.21, and it 'is . 1 1- S1 . cot7(y-C) 3.1). - '.F(C) 1:: .1..4 -0 (y) . , dy ..-. . 27 cos7-S Y S2 F(Y)P. -Y . , 2 1 1 b(Y)E(Y)cos7.- y cot-2(y-C)dy1 AO(n)cosh- { S r Re(---)

f

2-cosh fl + cos C )

AQ(n)cOs4

n

_im()F(

jig _{sinh Re(}

Jjdn

0 cosh

n

+ cos

FS

in)

'Z Is a mathematical abbreviation. The

complifcated'.funb-.Lions

appearing in-the last two integrals are very_

di

Efi'CUlt:

'

to

-develop ; The calculation can be performed only

numeri-cally, And in the present _paper. it is ' not . However,,

in 5 the integrals will be expanded asymptotically to treat a simple

'sp''eCial

example.

a

-where S2 ' , 2 1 cosh (-TO si.nh ( (3.2) X(4) 2frT. cos ,

-cosec-(y--,,r

0:(y5- 2

cos-ft.

_YE2, ,',F(y)

' 1

f

ab_(Y)EQ)cosec-2-

y-C)dy,'

t ' Al2(n) Re , [2cdSh,-TTi + cosh n +cos 4 . dy sinlY (3.3) E

_{- fb}

TTY-- (y) 2 dy. - (y)E(:)sin- y dy

k e

66)

d 1 cosh2

(-fn) .

Im 1

fr'ir3-1

Sinh(--n) 2 ni(frr7.9 1 )

Eq.. (3.1 ).1shoWt that,

unless

_{ie±'o, there is a}

singularity at t fr

In order that the Solution (3.1) is valid, the

,

1 2 1

-.2cos--'4,doSh (-n)

-17 1

(cosh n +cos Osinh---. ₂ n

+ 7

ACI(9) n 0 n+cos . -- 1 coSh

(-n)

1 \id 2 2

(cod,h n

+coS .c)sinh-fn

con-YS1 coslY

2 1

(3.N.) .

f

(y) dy

f

ak(y)E(Y)cos,y dy

b F

b 2

mu4t .1?0 Satisfied. ThiS, formula is when the ,real

.constant K of Ref. /19, p 151/'1S- determined. - The two

-equations ,defining : K satisfy boundary '.'condition

Together 'Cgs. (3.3) and (3;41) lead to

ST' , 2 -11 °b 177-77 dy

-f

b

(y)E(y)e,

S2 . . ,

LiPapding:

Eq. (3.2) in obseryi -(3.5): result

f

AS/(r) )cosh n Re(FT13-77)dn

2 ' 0

f

2,(n)cosh n

1417-7

-2 ln.

)dn

r

1 0

-+ f All(n)cosh

0

f

.60(n)cosh-2 0. n[i 28 sinh cosh n--1 sinh n cosh n-m(YtTiTY)

.1(411-711)idn'-series in exp(iC), and

.3 .3 s 1 AY _b dYE())/lb(1)e F e e -17y. , b ' -1(S2

11

1 coshn(AS2(n)Re(--7-).--t) PO(n)lir(=c))[-1-2coshn] , 2 rkin) , 2,1 , cosh

(11-cos Ike )

d is _r(4)(n)Re( _

AO(n)Im(Fk),-))[cos e- 2f3-1+4cos-h2nin and

(3.8) N

rYS1 eb(Y) -i4),.

-in -i4y

.2C. 2 . I Che. F775-:( cos 1..t e e +e - ) YS2 ' .3 .5 --17Y -123

-1.re

+ f

dyE(y)C2b(y)(-cos 1.1 e e +e ") b dn, _{)I tc.-370-1 .ACI(n.)Re.} co`sh12(ri) 1 . 1 ,

-COS e213 -1+14coSii n + 2cosh'n (-1+cos e 13)}

-i.

However, in 016 :linearized theory no use ..made of Eq.... (3,..8

- After a change of .variables the steady boundary condi-"

tion

(viii) - is

dw

.7., and observing Etis. (.2.4) and

(3.6:)--boundary

'condition _"(1j,*),.. ytelc.is . (3.10). 2a sin a

--9-

+ iae .1, ' 21T .

AdditOniel

constraints result from .the

fact

that the

velocity :` potential pust be continuous-. - On the 'basis of Fig (5)and Eqs ( 2. 2.ii ) and (2.5) one,.obtains.,

...

.(3.11i) + c :G = 1 0 . 2 U 7

. a

+ cos (y

2atcosa -cos.(y - -)+( y -

+2a .)Sina.l.

+17r . Si y

.117f _{'cl4)-e*PtX(010:))v±.E.surface'}

where c the distance of the stagnatidn Joint

frorn

'the ,

half-chord, and .

(3.12)- LiwuJin a = 'MT Uc + 2a(-c6s(Y _+P4')

, y _y

+cg'y) (YS1- )sin

Eqs.(3.11) and (3.12) are the connection between the angular' positions in different co-ordihate frames.

4. STEADY CAVITATING FLOW

In this chapter the boundary conditions defined by Eqs. (2.20) are substituting in the solution formulae of Ch. 3.

Linearizing Eqs. (2.211) and (2.3) the cavity length could be estimated from

(4.1) 4a0 = u/T47071 1i.

According to experimental observations /17, p. 440/ for reference cavity length one should choose not the mean cavity length but the shortest length in the cycle, i.e.

(4.2) 4a0 = 1.4/14-a0+10 1 L.

The steady part of Eq. (2.4) is

(4.3)-

0_C1,0 )C2,0 = 47aosin ayo.

4

Substituting Eq. (3.3) in Eqs. (3.2) and (3.10) respectively yields YS1 sinly 2 (4.4) )0.12.0 -

F(0

2,

f

1

f

obo(y) dy cos- c y 2 S2 -

f

(y)E(y)sin7ydyi b b0

S11

cosec-(y-) ebo(Y) F(12() dy YS2 I 21)0 (y)E(y)cosec (y-Odyl, and ia y0 . (4.5) 1 e (Q +1[4) -4) ]) 0 C1,0 C2,0 2na0

- The YST . 3

-1-y

..r.' -i-,.,

f

0

(y)-

_{dy + f cr':(y)E(Y)e.}

`.-. d b0 F-77)- b0 Y b S2 y

S1-y

1 n2

f:

9

(y)-4-7-(T), dy +

XS2 y_w0

ia

_y0

a0

steady part Of

yS

04.6)

Eq... (3.1)

cos- y

₂

177- "

,. 32 (y )E('-y

take she form

. ,

The integrations included in the above 'forillulae ,result- in:

le'ngthy, expressions

.

They are given in Appendix 'A 'for th'e

flat plate case.

The constraints of the system,

( 3.1,2Y and:

),

are now

.(L4:7)

Liird sin

_'y0

= rUc. + 2a0{7co's(ysi +

cps (ys2+ayo.) + (y

+f:-0150(y-)E(yfeos-4. y dy.

2a feosa

ta)+(y +2a .:)sina

1 a

10

yO, S2

yu.

.yu

:U f clexp(X(4(W)))'

_{we e.}

surface'-2a I

_a

_{-,-y0 S1}+a_y0

)4(y

_S1-

+2a lsirks

_y0 (:.;;8ii)" c(;b-)e.

.f do extili(

(1'1)

-)wetted.surface*

0

1..-changc.of varidbles and the el i min;it ion of

2a1S1

_{u r}

0 "

t.1

(sin[y+ot

,i+sin- a

c;'),dy.:.

qtantitatieS-14)C1,0C2,0)-' /1'

a0

'4y0' aPF1 YwO there exist 1 seVeTI "ec19.1t4-0P.s ( 4 ) , (,Lf.'3) , :5) ('real and

imaginary parts (4.0), -(4:3 ) , and (4..9). There are two.

possible ways of assigning, values to. the,width of the steady wake. They are -diScussed by MiChel /11, p. 226/, and,

Tulin.

/15, p. 20/, respectively. For the .present,'ywo is considered a parameter only.

The 'solution Of the equation :System. prOCeedS

E4s.;(.4.7) and .('4..9.), together result

in

(4:10) (27r-ysi )sin a + cos(y

-c°a(YS24.ay0,._ and (4.12), YS2 Y -r

e

(.0-PLY+ay +sin q

The 'substitution of the

special

values oft]

e

ekample under - Contiderdtion

into

Eqs. (4,6) and (4.5) yields

(4.11)1 ln(1+a0)sin (3- a cos (3 - 2arctan cos (3

n ta0 v a0} sin - 0 ,

f a0 ia -y0 . (Q-u+14na-sin ayo )-27ra, , 7 0

io

-ie (2arCtan./f-77-

_G e -

/-cos

u e(10

,

fireo(ft.o) f vrfr7-77 + 1nVt("-7-Tra°1(6.cos

.+

( cos.p)e -k'f" 1:Ef1-a.0: - ta0

ra0

f

- + _(4+o, mop*id73 0 _

ra.0

.

34 + if-2aGsin8-2arctanir----sin - ln(1+a0)cos 0 i-a0 f YwO iay0 1n1 tm° cos 13]) -Ifta0+Vf-a0 a0

The quantity QD (and simultaneously ywo) can be eliminated from the last equation, i.e.

(4.13) [2aG][ (-cos p sin213+sin a cos (3)sin ay0

+ (cos P cos2f3-sin2 a)cos ay0]

ii---[

I-a0 ][2(1-cos p)(sin28 sin

aO

-cos20 cos a )]

Y YO

l+fi-a0

[ 2arctaniF71---]_{v a0} I(-cos p sin213+sin a cos (3)sin a YO f

a ] + [ ta0ir;:--v a0 ]

y0 2

fi-a0-(fta0

[2(1+cos 11)(cos2a sin ay +sin20 cos ay0)]

f

-VT----+ [in{ ta° i-a°11I({6 -cosp -cos201+cos2_{p'sin lay0} f

tao i-a0

( {6-cos P}{-sin2(3}-cos 0 sin a)cos ay°]

+ [ln(1+a0 )][(cos p(-cos213)+cos2 a)sin

ay0

(cos p(-sin28)-sin 0 cos a)cos a10] = -4sin a10. At this stage one has three equations to determine the three

anglesY

, ay0

v and . To evaluate the integral in Eq.

S2

(4.10)

sur,0 is obtained from Eqs. (3.2) and (3.3) as Q

1S2 .{sin[y+ay0]+sin F(y)fysur,0 1 ln(1+a ) (4.14) asur,0 1 ₂ 0 cos( -y) 2 sin-(y +2c* . 1 )sinT(y-yS2) + 2 artanh . 1 . 1

sin -(-2a -y )sin-(y -y)

2 y0 S2 2 S1 for y <

-2ayO'

F(Y)fysur,0 1 ln(1+a) sur,0 1 2

cos(r)

1 . 1

sin-(yS1 +2ay02 )sin-(Y-YS2)2

+ 2 arcoth

1 . 1

sin7(-2ay0-yS2)sin-2(yS1 -y)

for y > -2ay0,

where

= aGsin + arctan 1/Y7F;-0. sin 0 (4.15) fysur,0 -11----+

in

f ta0 1-a01 1 cos

a+ 7

ln(1+a0)cos 0. 2 f

+/F7=7-ta0 v a0

Expanding the integrand as a Taylor series leads to

Ysi

a

0

(4.16)

f

e sur, (sin[y+ay0+sin ayo)dy

-2(a10]

_f

1f1+ui.(1+u)/F\f

riiir=A1 u

k1-u)k

1 ) Ysur,01

1S2 0 cos -y2 YS1 y0 ' _J }d, r 1

f1+vi11+v\

11-v k1-v)

1S2

(11/2_

1 )rysur,Olfsin[Y+ay0 ]+sin ayoldy,

cos y

wh'ere

. . ,

Wheni performing the detailed

calculati-onS

u and v are

:

found - to. be "suitable new variables:,. ',The reS-ultiLg.

cxbreSSion

e

is very lengthy, and it is given in Appendix B. 36 . 1 ) Ysin-(1, 2L. '1

0.18i) [1611+o p)A-L

2 a -.0 . +7.41---Llicos2(190-sin2(11j) -iTIH7171.' - - .

21[In6+a

M.-cos-10

_[G-a 1(2sin-10)1

2 , - 2

' 2

--1: 541.Pi/TETI-Tr] + 63[2(2-cos

3sin iw6i75-771

'an:1 on pcgTeCting second order contributions,

ln(11-a0)'Nsin-0-+ u1-24i.1] - 6

2 G

- 1

' .VT1777,

:

are i

trod

uced /18; p. 133/ . In t tie co-fit inuation it mist,:

1?e,..a,psumerd tha < 1 and 62 <, _{Etr.;' (4.11)}

rakes

.form :

solution :through iteration new io facilitate the

Similarly, after the substitution of the new variables, Eqs. (4.13) and (4.10) are

1 1

(4.19) 4(62_i2)) + 2aG[cos2 p +-cos

II--2]

2 2

1

+ ln(1+cr0)[sin p cos p+ - sinII]

2

1 1

+ 6[2)/-2 _{2cos2 - -cos I,- -}

2 2 2

1 r 1

-

217-47-174 t-2sin P cos p + (6 - 7)sinP 1

+ aG[0(62) + 0002)] + ln(1+00)[0(62)] + 0(53

and

(4.20) r(2sin2p) = (1-cos p) + 0(6) + 0(62) + 0(02).

Thus the approximate steady solution (Eqs. (4.18)...(4.20)) has been reduced to a suitable form for iteration. _{For long} cavities

(00

<< 1) the derivation would be notably shorter.

5.

UNSTEADY CAVITATING FLOW

In.thisiohapter the steady state solution is subtracted from he complete solution given in Ch 3.:. The treatment of the unsteady part is -Very simple, provided their second. -..Order contributions are neglected;

At first, the .functions AO

':-and. tar:

needbe developed.

The-deriVation of AOw follows Ref. /17913P:

452...454/--vei:y"cloSely.

"j_et A

be the rate at whiCh theunteady component

ofcaVity volume is passing'intdcthe wake.'

Then.

-- f

u dx, as

4/4x,

fl,whereyc:is the unsteady_ingrementtO::the:,dis,place-ment thickness at

C1 (see Fig 5) Thus E.q:,

)pecOmes

!

(5. Udy )

Let-,.V(y denote the cavitY:y0lame., Tbe_consel^Vation of -volume requires that -17/ The hypothesis to

be adopted is that a functiohal_relation!exists, between.:

.

-V.(T 42) and the pressure increment

'c

a

pcwekpfjW,ctil. The time delay,is:-introduced on the

-reaSbnable.assumption that V 'cannot respond to

pressure 'change immediately, Wait for

reaction ,ofthe re-entrant jeta:t'iC id Modify the flux'

of

bubbles

into

the

wake.-, .

The problem is to find' the relation between- V(t--1:2)

(5.6) (5.3)

T('14.2) = -Quc(T1+2) - k _P(rU)2{pC(T1)-pc0}

where k is an empirical constant. Hence

(5.4) k

{D (T

)

-D0

-c 1 .c P(rU) 14-2 = (0

-0

)

f

u(1,

2 C1,0 S1,0 1' -,7- 1'

and on transforming this with respect to

Ti,

one obtains

(5.5) k 2{U9.1 L(Pc(T1)-Pc0 P(rU)2 (.C1,0-(PS1,0) 2 2p

_rpe

_"u),

2 p C r P*EUT and i(Pc(T1)-Pc0) pew

-41-

_P

In the above u denotes U(x)u1 (1 x)" and L refers to the two-sided Laplace transform /3/.

As u is generated at C1' and must have a phase fixed in relation to the waves incident on

C1 from the cavity and independent of wc, it is necessary to write jw,(t1-2)

(57)

u = u e C -C and hence -j 2w (5.8)

_L(0cC

) =ue

cP

P-34)C

Once. u

ie

known, the 2

.14 )

" where

r(-1)

jw 1J cg T UT _{e C} e u 1

original, function can. be deduced

k2[141i5 ljw _ +72},,

[ ---2-:- .2i: . _....

P(rU) (ED - - 9

.C1,,,07 sf; 9) : ..

2sinh

(7) cos alro

t

s2n+-jw 12r

f, E f _ ccosayo+cosy S1C°Srly0SinS1inCL 0 (1T.YS ) [2r2f +21. 22n s 2n

Due to the;a:ingularity, at the end of he

caVitY,'

the

valUes

of

-

.a.-ithe

ipoint are

unknown'

quantit

'They:are

=

jw T (5.".11)("2

n '

1 .

wcv

,ci

-e..j14 T-,c. 1 wC2 =C2 j'cocTf.0.1214,?i).2 = e 1,

W220_¼23

-C2.

111.47 ricl.; S102 are determined by the boundary conditions

,inRef':. i'li./

equivalent terms do not apiDe'ar:....at4.1i

they

.cancel each other '.because Of .symmetrY.

'

l'he difference A(/' is

obtained' in7 a.:,.

milar Way.. .

' w :

strength is given by jw Tl jw c

[-y2+r2]

(5.12)wl(n'T1) =

e C -01 jwcTla -jwr2fs2n, = e -C1e jW T -jw r2f

(5.13)w2(n'1) = e

c 1 C2e c s2n2 -where . 2 1 2sinh (7n)cosayo s2n2

cosay0+cosy31 cosa_y0-sinyS1 sina +(n-y )sina

y0 S1 10

{cosay0+cosyS2cosay0 -sinyS2sinay0+(-n-yS2)sinay0}2 and

(5.14) _{aw(n,ti) = Quo(n,T,I) - Qw2(n,T1).}

Now the unsteady boundary conditions, defined by Eqs. (2.21), can be written down in greater detail as

(5.15)

a

--CU WCT1

(i)

2bw(y) = _{2 1+a}

0

(ii) Obw = n in -2a < y < ayo, n = 0,

Obw = 0

in -n < y <

-2a1, -2 YO< y <

n,

n = 0, where _{ay = ay0 + ceywexp(jwcti} ,

, 2, . r2{1.11013 fjw ir J] jw T., -2r2cuTf Aew(n) cw c UT _e C _{e -s2ne-jwcf22n} 2 p(rU) 4C1,0-(I)S1,0)

_w

wcrlia

eAr2f

c s2n_sl _{e-jWcr2fs2n2]}

aw(n) =e

-C1 _-C2 2

in

-"Y<YS2' YSl<Y<n'

or -(vi) " and ... (5.18). respectively, 0_a

(n)

= AO e_-a

jetl

jw -(vii),

w(n)

.A0 e

AnalogodslY: to Eqs. (I.?)

iwc:T1 ,..(5,1 6) a +."a.,

a

_, ^1.13 jv T

.

c: .. , .-- + a e y0 -yw a,

.where:-and a

are the UnknOwn unsteady cOmponentS:. _ _ ,

_ . Then .the,).unknown :quantities! .(k;'.

cut '.:1/to'.7n2C1 a and'

). are to be determined:: Both the steady and

the

-.Unsteady boundary ,conditions are subsiitUtedin-qqs.

) , (.3 1 1 ). and ' (.3 . 1 21. After the subtractiOn of 'the.

, , . , . .

,etc:ady-

pari,

the. unsteady contributions of.,:tqs. (3.10) and

-'Jr(

3:12)

are left, and they are

uc

]

2

iT747

0

-simys2+ay0)

+ (-21T+y

The substitution of Eq. (5..18). in ( 3.11 )=-Yields,'

,

respec tdvely..

(.5;1:6 ) IsinKY - .siray +a ).,4 (.. 27+y - - )bosa

Ysi

/T0

TE- f sur,0(coely+a, yoj+cos a10)dyl YS2 1 . Ew _r Lcos(yS1+ay0) - cosO( +a ) - 2 1+a S2 y0 0

(-2"YS1 -Ys2)sin ayo]

Appendix C contains some of the formulae needed in the detailed calculations.

For _{y >-2aYO -Osw}f is obtained in the same way as is shown in Eq. (4.16). Substituting Appendix C into Eqs. (3.4) and (3.7) leads to (5.21) (a e waT1) -271-cos aYO -Yw 1

Ain(aYO

+lyS1 )sin(-ay0--fyS2)2

jw T

(e

a C 1) nsin B / -41) +

f

AS1 (n)cosh.712-n Re(13.n)f . )dn

T

(1+a ) 0 0 where (5.20) /1-TETI fS-2sw + a + (1 + for y YS1

f

YS2 a ( fasw(sin[y+a10+sin 1 ayo)dy, (1+11\ (1+4F(y), 1

_u"

i

+cos8>1 k1 ysur,o cos-y 2 0 (1+u) F(Y) 2(1+a 0 0 -sin _a1 u F(y) Si"in(ay01-2-Y )sin(-ay0-21s2) ( d (1+u(a1)))

f)

ki-u) cosr ay ay0 Ysur,0kdaykl-u(ay)Il co1 s -y 2 < -2ay0.

44

dn

f

_{pO(n)cosh .21 n} .0 jW T _ (5.22) (a e c

1)

271 {cos(a -20) 1 1 y0

/sin(a +-y -)sin(-a --y

y0 2 S1 y0 2 S2

,

I:firsin(a_{y0 2}+11 )siii( -a -1y-A'--8tih3O'sin(a +5)

s1

- 2cos u'cos3

_cos(0

ct I:13) + i[gin(a -20)

y0 1

0+7YS1)sin(7c-1YD7

--+- 2cos ii sin36 cos(a

_+OA}

y0 3wcT1 ir cosy

(0

e )

--13+3.cos3_13)

(sin3 1+a°

ika

iwbT1

"

e ) Jr.coss.,

1+a'

- 314c 1 e ) 0 Yw 2.7sin a YO 1 1 s y02yS1

in-1(a +- )sin( -ay

2 0-7. S2 ) ,6,55(n) ! - f AO(n) 0 cosh-1nsinh₂ n cosh rii-1". 1 . cosh- n sinh n cosh n-1 AG( nMir(F73-54.1-))dr1 ,52(1)Im(yijitit):A!:-)]-8cos313

_sin(ay0

+$) , f cosh-Tn.(-1-2cosh -Ftan) 1 w!

+If

4- (1. 2C-oqn 0

sinh.1

n: n

)Re.(mym

Formulae (5.21) and (5.22) contain six equations to deter-mine the six unknowns k, _{CUTS 0-1C' and}

-2C' The integ-rations can only be performed numerically, and this is not done in this preliminary investigation.

The quantities k and styr might form a basis for estimating the stability of the cavity. A comparison with experimental results could prove that there exist limiting values of these two parameters, beyOng_which the cavity will collapse. Another approach to the- stability of a

cavity is given in Ref. /3/.

However, the case. when we approachesi infinity, can be attacked easily. The integrals are expanded asymptoti-cally, and only the first terms are considered.

Eq.

(5.21)

and the second part of

Eq.

(5A2) take the form

--2ncos ay0

(5,231) + a usin0

Ye / _-

w TTTY-7

₀

isin(a )sin(-a --y )

y0 2 S1 y0 2 S2 jwcr2fs2n

+4

coshin[-C10 e-2 0 -jecF2fs2n2]. ( 1 0 e -- -C2 xe T7-377)dn 1 k2fU2}pcwf jwc+cutir2 )

f

cosh7n[ -2 0

P(rU) -(0-0

C1 0 S1 0 - r e f -jW f

e

UT s2ne c -22n T ( 1 '111

777

0, 2u

(5.23ii) a

_Ye,

1 in(a --213)

1 l' Y

isin(a0

+ly2 S1)si. y0n(-a -.yS2 )z

1 1

../..,.purts an

+ 2cos u sin3I3 cos (a +8)} ncos cos313

-w (1+a ) 2nsin a =_.ncos 0 YO yw

, YOfww

+

i

( + a +11St )sin(-ay0 -lyS2 )2 2 . . ,2,1 , 2

f

2cosh .cosh %.70./ - , -C1 - e 0 sinh -where -jw c. 0 171-07..

.(5.25)

k = 0, and .(5.26) {f }p p.3.4 c 1+cos 1r 3 1j' 1-= -rC2. 2r 2j* , In the above f

-that .,depend only on the

46 s2n21- ( -1 ) Im

TurTy

+e

./7)- icos207.cos c--VT VT-11,1{1"+cosy0' r IPCW{iwc+cutir211e-jwcf2 rk2{Ult P(i-4-1). (SC1;071,0)

Applying. ,Tq . (.6.1,5) of Ref.

/2, 'rp.

220/ apd:°11'. ./"Vj:11g.

1 g

= {Q.:--11+cosy

]12-3 - irr .S2

-is a known funCtion containin_terms

If one wants to discuss Q and . 0 separately, .a higher )

- :Order expansion of the integrals or

numericapt,bach:-.

-..Wbuld be' necessary.

-The fact. that 1( =0,

Seerils:t

p be idhysi.Cally. feasible,

wnen w6 is approaching

infinity..

- -.

Observing

t4. (2.7)

the pressure 'difference, becomes DRAG AND LIFT

epressureson the wetted surface, and on the

]

ofithe'joiI are Obtained from

.2 alOsur-P.Sur +-7,Pqsur P arid 4848 1 2

a0c

1-7

Pqc P 3t poo

7

P P sur 2 30sur - _ = r_ 2 2 '

-7

U2

,.--Thelorce coefficient per unit length of the

foil

ia (6.3) t C =

T-1'7 f

(1361a 2 Substituting; equation sun

2Aisin(y+a )+sin a = de,

Y Y

Eq,,i(6-.2) into Eq. (6.3') gives

2a

Ysi

'acti

= (1ia Y=-

f 2 -sur.D G cU

ysa

a

(I+a).esu r-e sur)fsin(Yta.)+sin

4 My'

Y y

within the linearized approximation,

3tdc-T1 + (y+2ay3)cos a,,]}= -2a . .sur. + -e ; 34sur . e jw

cti, {2a [cos- a, ,cooy+a, )

c -7=4.1 Yb TO .

ca-vits,

)sin a 2a a [sin

a9

4-sin(y+ )

Sin .a.lio)d (ces[y+a ...(sihEY4a Y dO/dt YS2, +_ sin a )dy.

.The.steudy component of the force' coeffrcient,is,

YB'r Q."

(6.3) '-iCD0 = (1-ia )

G - -cU ([14.061e P1/7.0

'---11.sur;0)

To evaluate Eq.-.(6.7) both

exp.(c sur0) and, expC--il

-,

7 are expanded as Taylor SerieS. The resulting

Idris:thy:

-expressions are giVen in Appendix B. .

-' Separating the unsteady bontribution:ofEq. _{6.5) yields}

iC = e c' 1 '63 EC -iC ] Lw Dw a L 0 abbreviation. . y 71,:c_11

f

-ell'''. 0 '

2 dOidt,sin[y+a,0

.S1

u

+. a.. _{((l+a. )6} r,0_ . . -Y.B2 YS1

l+coS

ayo: ).dy + _{j (,(1+:30)1a,sw-1(_)Qsw,)'} 1S2. +iiri ay0)dyl, Sur.,0). .Where_ 1(-1)ilSili ' ._s_'en-expression similar, to. . (5.2_0, ..

-At,tbisHinitial

stage.

of the exploratory survey, a detaii-ed.

... calculation of Eq. .(..6'.0) _{would not increasejhe'physical}

insight.-intOthe problem. Thus the integrationS are not 4eyelOped futher.,

CONCLUSION

Tie .!purpose of the riresent preliminary study was to

,_

..investigate the feasibility of a new way of formulating the boundary .conditions of

a

,cavitating foil in unsteady flow This aim has

beew,SucdSS

f u 11k. accomplished . In the

.case of !.,,,a1flat plate. the Solution obtained contains in the

main_ simple, but,lengthY algebra. The:complete mathematical

.eValUatiOn

of

the approadh

"iS

only possible by applying

numerical meth"Ods

-tghen, calculating .cavity induced pressures on near-by':

surfaceg, one of the key problems is to estimate floW conditions under which, the cavity bubble is likely to collapsS The present approach might be applied to this. problem as follows. It could well prove that some limiting values of the parameters and k exist, beyond which

- the cavity bubble will T8ollai5Se.

In the future this concept of formulating the boundary. conditions is to be 4-plied to several other Unsteady cavity

1

problemsr. The theme- Of.oneof the following papers will be

a. .cavitating hydrofoil in a,base flow thai',COntainS,:minor disturbances.

ACKNOWLEDGEMENTS

The author expresses his thanks to Professor V. Kosti-lainen for making available the facilities of the Ship

Laboratory of Helsinki University of Technology. Also my thanks are due to Miss E. Heap for correcting the English text of the manuscript, Mrs. I. Lauksio and Mrs. I. Halenius for the typing, and Mr. P. Tuovinen for drawing the figures. The financial support of the Academy of Finland has made the present study possible.

REFERENCES

_

Mechanics 71 (1975) 2, pp.,:9,39.:..35.9: . ' Gradshteyn I S & .Ryzhik

Series and Products. New York, Academic Press, 1965. -1086 pp.

Kelly H.R. , An Extension of the Woods Theory for

,

.Unsteady.

Cavity

Flows. Journal of Basic Engineering

89 (1967) 4, pp. 789...806.

8 ' Knapp R & Daily J.W. & Hammitt F.G. Cavitatioh.

. New York; McGraw-Hill Book Company; 1970. '577 pp.

9 Kochin. N.E. & Kibel I.A.

& ,Roze N.V.,

-Theoretical

HydrOmechanics. London, In Cersc eno e Pub 1 shers ,, 1964.

577 pp

Larocic ,B.E. Id_ Street R.L. , A Non-Linear Solution for ,a Fully Cavitating Hydrofoil Beneath a Free Surface.

Journal of Ship Research 11 (1967) 2, pp. -131 :..139.

.11. .Michei, j .-M. , _{Wakes of Developed Cairities. Journal}

of Ship' Research' 21 (1977) 4, pp.. 225.. :238.,

1 Abramowitz H. _{Stegun Handbook Of Mathematical}

-Fuhctions.. New York, Dover -publications, 1970: 1046 't5p.

,

Handelsman Asymptotic Expansiona

New York, Holt ,'Rinehart and Winston,

4.26, pp_

-Brennen C 'Cavity Surface 'We've Patterns and General Appearance. Jdurnal of Fluid MedhahidS 44 . (1-970) ,

_Tp..33 49_

_

,ffuruy4 0,, Nonlinear Calculation of Arbitrary Shaped

'

SUppreayitating. Hydrofoils' Near' al,Free 'Surface,.. l'o-drhal.

of FlUid.Medhanics, 68 t19.75/. pp. 21.:,40..

Furuya,. O.,.

.Three-Dimensional Theory on SupercavitaiiPB Hydrofoils Near a Free SurfaCe. Journal of Fluid:_{_}.

12 Silberman E. & Song C.S., Instability of Ventilated Cavities. Journal of Ship Research 5 (1962) 1, pp. 13...33.

13 Song C.S., Pulsation of Ventilated Cavities. Journal of Ship Research 5 (1962) 4, pp. 8...20.

14 Street R.L. & Larock B.E., Two Models for Cavity Flow -A Theoretical Summary and -Applications. Journal of

Basic Engineering 90 (1968) 2, pp. 269...274.

15 Tulin M.P., Supercavitating Flows - Small Perturbation Theory. Journal of Ship Research 7 (1964) 3, pp.

16...37.

16 van der Pol B. & Bremer H., Operational Calculus Based on the Two-Sided Laplace Integral. _Cambridge, Cambridge University Press, 1950. 409 pp.

17 Woods L.C., On the Instability of Ventilated Cavities. Journal of Fluid Mechanics 26 (1966) 3, pp. 437...457.

18 Woods L.C., On the Theory of Growing Cavities behind Hydrofoils. Journal of Fluid Mechanics 19 (1964) 1, pp. 124...136.

19 Woods L.C., The Theory of Subsonic Plane Flow. Cambridge, Cambridge University Press, 1961. 594 pp.

20 _{Woods L.C., Two-Dimensional Flow of a Compressible} Fluid with Infinite Wakes. _{Proceedings of the Royal} Society of London, Series A. 227 (1955) 1170, pp.

367...386.

21 Woods L.C., Unsteady Plane Flow Past Curved Obstacles with Infinite Wakes. Proceedings of the Royal Society, Series A. 229 (1955) 1177, pp. 152...180.

piPP,EN.OIX A Integrals for evaluating x , Taking _ and' 9LDO(Y"'-'3!.G 4 Tr14("2.y0-)

the integrations

in

Eqs. and (.4:5)

7111(1+ao.)n cbs 8, 2 'EA:3)

_f

c(y)E(y)e.dy =

[ (A.11.) ibo(y) = - _ln(1+00), ' 1 (A.2)

f

0(y)E(y)sin - y dy = b 2 1r-;(1+a ie-i0, .3 2 S.

f

0(y)E(y)e

dy =o

_b

12*

cos

II ie...

(A.5) 0 (v) 2 b(1.

'I

F(y) YS2 -1n(1+b0 ) irCOS

-[1?-arctanVT;f703[-.2sin

Ciao

/175)]

cos

I/14-

/

-7a0 f tan(-a ' Y )sin an( -a O)cos

a

-Y dy =

[Ira

are

_as

3

2,1

,(tan(-a )sina+cos0)2

1

fi-ao

= [-cos2

(7p) +

sin t-p)

2

tan(-ay0)cos B-sina)

.1

YS1 -3.7y

(A.6)

_f

_{eb0(Y) eF(y) dY = Ina0][-2cos B} + i2sin a] 1S2 In arctanifrol[-2cos a 4- i2sin

a]

[Tr 1nffta0

I

;0HC:7

(.

sin 0 + icos 0]

kfta0

+ =

[naGII

-2e -16 + In arctamq----/-a0][e-2 -i6]

+ [n lnffta0

_{7 v5ii-°_1]}

Ile 16],

fta0

+ rf

ra0

.3 yS1

-1-y

2 (A.7) I

_0b0(y)r(y)

dY = [naG][-2cospe-316] (A.8) YS2

[n arctaniF7:p -2cos p e-3ia]

[n lnfta0 -

fr7c71,r

i(io-cos P)ie-3i6]

fta0

+ 1-m0

[Tr /F7-7Z

[2(1-cos p)e-3ia] I 1+fl-a0 ff----[ f ta0 rat)

I[

2(1+cos p)ie -31a],

fra0-(fta0)2

F10 r

0

(v,

E(Y)

1

di = [- ln(1+a )]

2n ' bO''' 7 0 '

The two integrals

1S1

(g.1)

..e

a

+sin 6 -sur,0

(sinEyfq1+iti

: .

appear.-frequently

,in-the,

derivations. 'When, Sxp(O . ) 3.-s

. . , . sur,0

-expanded as Taylor series the expressions tor the

. ,

_ .

. . .

abbreviations

_{Isubc-1,ipt.,}

are-asfollows-. . _

_.,

. . .

-

. , °( B. 3)

{la

sin...(

+ 0

a

)cos

_{yS arc}

tan

_f

+cos P

1. 2 '1 sin '

1(fSSsin

11 +

4siny(-2ay

)c'os1

2 arctan, f

_{COS II/}

SS /.141/,fss2sin;41 qCOS12-1 1.1A : ) + Ti S11171P +T.s 1 1-=if--2-sin-P-4-f -SS' --Z, SS

8F-(-41'.y {-COsly

rfsszcosr 1. 2cos: Sk 1 Cl4f '2cosu+f .cos -/ SS 2 SI

PPSir4(, ayo-IspCO3;

ys

1 j

)]

-..1+}{1 sin _ 2 1 [cos- P 2 cr-1-sin2(1p) - SS 2

in;.

P

21

2 cos (-2p ) _ 58

("srs,

-SS 2cos )J cos cosTys2 . 1 1 1 2sin--(y a )cos-2 S1 +2 y0 2 1S2.1 1. Y -yS2)cos-y.2 S1 i - 1. ". 2cos P 1 - } { co s-Ty , 5 .. - . Co's--)% .,.2 S1 1 1+,/fss p

+fss

cos- I/ ln( -- 2

sin2

(-2p ) 1 -P/T3;. 2sin- P_{7 +fSS} . 1 sin-2 11 2 ,er_{2 arctan} ,

n-fss

cos 2 f 4 +

(-sinl-p (1-f )+a--- 2cos2(1.))

, SS 2 SS SS 2 -2 't cos -2. (1_+fss2cos fSS) _1

+

-}{i

-sn-(y

1 1 a

)cs-

1}{cos-2-ysi

+ 2 2 S1+2 y o yS2 1 2cosP2 co s-,y z S2 1 ln( SS z 2sinw p

+1)

f

SS"IcS7:

'2si44

+1.. + arctan{ _

.1-12Co4p

SS -( -sin-p-( f -1 )+1/T-2cos - 2 SS SS -,

cos-- p (14

f 2cos p +

f

)-2 SS SS

.

-sin -(-2aYO -yS2)coslyS12 Ifcos-/2 .

2 S2 . 2 sin p cos-,y z S1 + fSS(-1-2cos211 -cos p

co1

s-2 YSs-2)1 cos-2 YS1 -cos 11 +f

(1-2cos211)

SS

farctan(sssiin ti pi\ + sin P

1+fSS 2cos P + f2 SS 1 1 1 Isin7(Ysi+2ayo)cosy ys2licos-Tysi flcos-Y cos p 2 S2 sin2p 1

cosesi

fccsin p 1 (

""

cos 1.11 + ---(-1-2cos2p farctan1+f -cos p SS cosyyS2 SS fSS(1-2cos2p -fSS cos

0)1

+ sin II

f2

+fSS 2cos + 1 SS 1 cos7Pfi + flf4F(-2a_y0)co 1 sTYS1 (,1+/F--2sinlp+fSS 2 SSN . 1 1 in

_I+

sin-2141+ co ----7-T

(7p))

s \1-1F--2sinlu+f / SS 2- SS

rl-f

/r--(-cos +f _I-1+sin20)1 (arctani SS 1.1.TrNi. SS P SS

lri---2cos1-4.1_{SS 2}

2)

sin P (1+fSS 2cos1-1 + f2 )

SS _ +1-114F(-2c )cosl ircosTp/ 2

kl

1 +1 YO 2YS2J1 1 ) 4sin'(7p)

)1

, . 4 160 tf SS+ s 1+ 1 1.1( ink ' 2 u 1 4cos ' SS SS 2 VT:77-( - f+.2 .61"i1-21.c4:f co s 7p

Yr

t;(,.aretan[

140

+ SS ;irTs2cos71.1-1 + fss) , (B. 5.). {I1"() =. [

22.

sin..11,.sn ' 1 : :: '-._. -Ysui-, . Ct"-0 2 1 1 +i

{cos--

ln( 2 2 s .4- 2',-:-i1.i(arctan ,. 2 [1-f SS , , _... rf---2coa SS , .

f

liS

2 sini

fir

+ arctan[ ...

2 2 ..., _41T.3-2(..,' os7u ,...+cost.P lil - - SS

+I 1

(1 +),/,T2si.ril--11 + .- f

)1

i: 1 2

1 -Cf-;2sin7 PL-J

! fSS -f s , . +

F ( -2}(1-2cos

-+fs2

s

,

Co;

?(:13.0 1 +,/. -2s , 2 .1 ln( SS. 'SS) `.1./T"2cos-211 1 -.)/T-;2, sin711, ' 1-f SS, /17'..."cos

,ss -.7

p :'-7772cos14

t

1

2 [ - sin2 (

r )

(1-f s ) VT-2co s 2 ( ) in-2-1p

SS 2. 2 1 + fSS 2cos p + f SS + f-F(-2ay0)/(-2cosP+fSS1)F--2

_.1

ss sinn

1 2 ln( 1 1411---2sin-p 4f sin (.2-p ) . SS 2 SS) 1-14--2sini.p+f ir--coslu SS 2 SS SS 2-f -1 [ SS

+n

(arctan /7-2cos-u]_SS 41 2-1 (wp)(-1+fSS-z )+VI----2cos2(1P)sin.1P)SS 2 2 2 1 + fSS 2cos U + -fSS sinp(1-f2 cos p F(-'2YOa )1farctan[ SS sin p 2 cosp(1+ _ +f2 fSs

cosp

S 2 sin p cos p (-1+f, ) ," + sini(y +2a ) 2

si

yo 1+fss2cos p + fs's f[2cos2(1p)sini p + 2 2

21

sin

(701/

1-1-fSS 1+ E\ +

[4s1n24p)co4

_Itarctanl - J% 2 La--2cos1pj 2) cos p SS 7 ir--2(-cosP+f_SS [-1+2sin2pl) 1 . 1 sin-(-2a -y ) 1 2 YO. S2 1 + fSS. 2cdsp + f2 SS

f[21

.1

12cos (71)s1n-2.p + 1

cos2 (TO] (1+1/7772.sini 2p +fSS\ ln 2sin7 p 1-/1-7-2sinlu+f -SS 2- S1S 2 1 , cos

(ri.

1 (1+iFTs-2sin-fSS\ 2 4-', lln 2sinl₂ 1-/F--2sinip+f SS 7 SS

APPENDIX C Integrals for evaluating )(to Taking (T = T1) a

jecT

1

(C.1)

_-

_{2 1+a} 0

the unsteady components are as follows:

.1

I abulE(y)sinv

dy = Stbal(2n cos

a)

b .

0 JUST

(n cos

a),

1+a0

1

f

nbwE(y)cosTy dy = Obw(-2w sin

_a)

a jw T (n sin 8), 1+ ao .3 .

-3ia

dy = Obe(-2n cos p ie ) (C.4)

_fbwE(y)e

2 0 jWCT

w

_{(n cos p ie} 1+00 -2a .1 y0

-1y

1 2 (C:5) u

f

rrTy e dy 2a

jwcT 2Tr(-cos _{ay0 + i[-sin ay0])}

= a e

Ye

2 clY .30 1 F(y) 2 (C.6) it

f

_dY -2a jwcT = a e

Ye

0/sin(a +-1 y )sin(-ay0-7yS2) y0 2 S1 1 2ff 1

1 1

fcos(a

_y0

-28)i1-4sin(a

_{y0 2 S1}

+-y

)sin(-a

_{y0-2 S2.}

'sin 38

sin(1ua..,n+13).

- 2cos

co's 38 Icbs(ay0+8)

+sin(a

-13 ) [1 -Ltsin(a

_{y0 2 S1}

+ly

)1

.

y0- 2.S2

co

Dependence of

t7c

at section CI=

..

MñdOfiirbIhOii-Wtufbatiati Volume