Re-Entrant Jet Modelling of Partial Cavity Flow on Two-Dimensional Hydrofoils

Re-Entrant Jet Modelling of Partial Cavity

Flow on Two-Dimensional Hydrofoils

Gert Kuiper MARIN and TU Deift

Jie Dang,

MARIN

Report 1151-P

Project Code; 963

7 - iO April 1998

Presented on the Third Symposium on Gavitatiön, Grenoble, France, Proceedings Volume i

Edited by f.M. Michel and H. Kato

TU Deift

Faculty of Mechanical Engineering and Marine Technology

Ship Hydromechanics Laboratory

Third International Symposium

on

Cavitation

APRIL 7-10, 1998

-

GRENOBLE, FRANCE

PROCEEDINGS

Volume i

Cavi

_tation

April 7-10, 1998

GRENOBLE,

_FRANCE

Volume 1

Edited by J.M. Michel

_{and H. Kato}

98

SCIENTIFIC COMMITTEE

H. Kato, Chairman,

_{University of Tokyo, Japan}

J.M. Michel, Chairman,

Centre National de la Recherche Scientfique, Grenoble, France

L. d'Agostino, University

_{of Pisa, italy}

F. Avellan, Ecole

_{Polytechnique Fédérale de Lausanne, Switzerland}

G. Bark, Chalmers

_{University, Got eborg, Sweden}

J.R. Blake, University

_{of Birmingham, UK}

M.L. Billet, The

_{Pennsylvania State University, USA}

C. Brennen, Calfornia

_{Iñstitufe of Technology, USA}

J. Field, Cavendish Laboratory, UK

D.H. Fruman, Ecole

_{Nationale Supérieure de Techniques Avancées, France}

B. Gindroz, Bassin

_{d'Essais des Carènes, France}

S. Kamiyama Tohoku

University, Japan

A. Karimi, Ecole

_{Polytechnique Fédérale de Lausanne, Switzerland}

S.A. Kinnas,

_{Massachusetts Institute of Technology,}

USA

G. Kuiper, Marin,

_{The Netherlands}

J.T. Lee, Korea Research _{Institute of Ship & Ocean}

Engineering,

Y. Matsumoto,

_{University of Tokyo, Japan}

S.S. Pan, China

_{Ship Scientific Research Center,}

R. Simoneau,

_{Hydro-Québec, Canada}

B. Stoffel, Technische

_{HOchschule Darmstadt, Germany}

LOCAL EXECUTIVE COMMITTEE

J.P. Franc, Chairman,

_{Centre National de la}

Recherche Scientifique, _{Grenoble, France}

A. Archer, Electricité

_{de France}

Briançon-Marjollet, Bassin

_{d'Essais des Carènes, France}

P. Dupont, Ecole

_{Polytechnique Fédérale de Lausanne, Switzerland}

F. Jousselljn, University

of Grenoble, France

T. Maître, University

_{of Grenoble, France}

Marchadier; University of

Grenoble1 France

J.M. Michel, Centre

_{National de la Recherche Scientifique, Grenoble, France}

J. Pauchet, Centre National

_{d'Etudes Spatiales, France}

C. Pellone, Centre

_{National de la Recherche Scientfique, Grenoble, France}

M. Riondet, Centre

_{National de la Recherche Scientifique, Grenoble, France}

Bubble dynamics in cavitation

Y. Matsumoto

University of Tokyo, Ja pan

CONTENT OF VOLUME i

INVITED LECTURES

3

Vorticity and pressure distributions in numerical simulations

of turbulent shear flows

9

M. Lesieur

Institut National Polytechnique de Grenoble, France

The prediction of unsteady sheet cavitation

19

S.A. Kinnas

University of Texas at Austin, U.S.A.

BUBBLE DYNAMICS

Details of asymmetric bubble collapse

39

C.D. Ohi, O. Lindau, W. Lauterborn, A. Philipp

Technische Hochschule Darmstadt, Germany

Bubble formation by a corona discharge in dielectric liquids

45

F. Aitken, F. Jomni, A. Den at

Laboratoire d'Electrostatique et de Matériaux Diélectriques, Grenoble, France

Interaction of a cavitation bubble with a curved rigid boundary

51

Y. Tornita

Hokkaido University of Education, Japan

J.R. Blake, P.B. Robinson

University of Birmingham, UK

Effects of bubble-bubble/bubble-wall interactions

on the nonlinear oscillations of bubbles

57

H. Takahira, A. Yasuda

Osaka Prefecture University, Ja pan

Non-spherical motion of two cavitation bubbles

produced with/without time delay near a boundary

63

K. Sato

Osaka University, Ja pan

Y. Tornita

Hokkaido University of Education, Japan

Spherical bubble dynamics : a model to predict red blood cell damage

by hydrodynamic cavitation

69

S. Chambers, R. Bartlett, S. Ceccio

University of Michigan, USA

Numerical simulation of bubble dynamics in vortex core by the BUBMAC method

75

KYan

China Ship Scientific Research Center

J.M. Michel

Laboratoire des Ecoulernents Géophysiques et Industriels, Grenoble, France

Prediction of Francis turbines efficiency alteration by travelling bubble cavitation

81

C. Arne, F. Avellan, P. Dupont

Ecole Polytechnique Fédérale de Lausanne, Switzerland

Interfacial tension in water at solid surfaces

87

N.A. Mortensen, A. Kühle, K.A. Morch

Technical University of Denmark, Lyngby

InteractiOn between cavitating btibbles and vortical structures

93

G.L. Chahine, R. Duraiswami

Dynafiow Inc., Fulton Maryland, USA

Linear stability of Blasius boundary layers containing gas-vapor bubbles

99

F. Burzagli

Cent rospazio, Pisa, Italy

d'Agostino

Università degli Studi di Pisa, Italy

NUCLEI AND INCEPTION

Effect of boundary layer tripping on the onset of cavitation in jets

109

G. Shridhar, J. Katz, O. Knio

The Johns Hopkins University, Baltimore, USA

An attempt to predict cavitation level in turbulent jets by a

statistical approach

117

Pauchet

GEC-ALSTHOM-ACB Grenoble, 'France

Influence of large-scale vortices on cavitation inception in steady, turbulent flows

.123

R. Leucker

HOCHTIEF Consult, Essen, Germany

Cavitation by macroturbulent pressure fluctuations in hydraulic jump stilling basins

129

RA; 'Lo pardo

Instituto Nacional del Agua y dei Ambiente, Argentina

On the cavitation of orifices in a pipe

(measurement of pressure reductions caused by vortices) ...

135

Ogura

Liquid overheat upon

139

V. Prisniakov

Dniepropetrovsk Sta te University, Ukraine

Nucleation and bubble dynamics in vortical flows

143

REA. Arndt,

St. Anthony Falls Hydraulic Laboratory, Minneapolis, USA

B.H. Mames

Lockheed-Martin TAS, Fort Worth, USA

The influence of nuclei content on .the inception of bubble and vortex cavitation

149

A.V. Chalov, V.P. Ilyin , Y.L. Levkovsky

Krylov Shipbuilding Research Institute, St.-Petersburg, Russia

Effects of dissolved gas on cavitation inception in free surface flows.

155

T. Bau r, J. Köngeter,. R. Leucker

Aachen University çf Technology, Germany

On the cavitation occuring at the bottom of an accelerated circular cylinder

161

H. Tanibayashi, K. Ogura, Y. Matsuura

Tamagawa LIn iversity, Tokyo, Ja pan

The experimental investigation of the cavitation inception

in the flow around NACA 16-018 hydrofoil

167

L. Wilczynski

Institute of Fluid Flow Machinery, Gdansk, Poland

Tip vortex cavitation.: effect of the hydrofoil fractionation

173

Viot, C. Jeanson, D.H. Fruman

Ecole Nationale Supérieure de Techniques Avancées, Palaisèau France

Method of similarity for cavitation

179

L. Espanet, A. Tekatilan, D. Barbier

CEA Cadarache, France

H. Gouin

Laboratoire de Modehsation en Mecani que et Thermodynamique Marseille France

Numerical and experimental study of cavitating vortices in turbulent wake

185

B. Belahadji, j.M. Michel

Laboratoire des Ecoulements Geophysiques et Industriels Grenoble France

HYDROFOILS AND PARTIAL CAVITIES

Three-dimensional characteristics of the cavities

formed on a twodimensional h.ydrofoil

191

Kawanami, H. Kato, H. Yamaguchi

Uniziersity of Tokyo, Japan

Flow in the closure region of closed partial attached cavitation

. 197

K.R. Laberteaux, S.L. Ceccio

University of Michigan, Ann Arbor, USA

Two-phase flow structure of cavitation:

experiment and modelling of unsteady effects

203

J.L. Reboud, B. Stutz, O Coutier

Laboratoire des Ecoulements Geophysiques et Industriels Grenoble France

Influence of cavity thickness and pressure gradient

on the unsteady behaviour of partial cavities

209

M Callenaere, J.P. Franc, J.M. Michel

Laboratoire des Ecoulenients Géophysiques et Industriels, Grenoble, France

Investigation of unstable cloud cavitation

215

T.M. Pham; F. Larrarte, D.H. Fruman

Ecole Nationale Supérieure de Techniques Avancées, Palaiseau, France

Evaporation rate at sheet cavity interface

221

Y. Kawatiami, H. Kato, H. Yarnaguchi, M. Maeda, F. Oshima, J. Doi

University of Tokyo, Ja pan

Cavitation inception and development on two dimensional hydrofoils

227

P. Dorange, JA. Astolfi, J.Y. Billard, D.H. Fruman

Ecole Navale, Brest, France

I. Cid Tomas

University of Valladólid, Spain

Oscillating partial cavity on elastic hydrofoil

233

Kovznskaya, E. Amromin

Krylov Shipbuilding Research Institute, St-Petersburg, Russia

Numerical simulation of 3D cavity behaviour

239

Maître, C. Pellone, E. Collard

Laboratoire des Ecoulements Géophysiques el Industriels, Grenoble, France

Partial sheet cavities prediction on a twisted elliptical pianform hydrofoil

using a fully 3-D approach

245

R. Hirschi, P. Dupont, F. Avellan

Ecole Polyt&inique Fédérale de Lausanne, Switzerland

PROPELLERS

Air injection as a means of reducing propeller cavitation induced ship vibration

253

j.W. English

Maritime Technology, UK

Correlation investigations for higher order pressure fluctuations

and noise fòr ship propellers

259

J. Friesch

Hamburg Ship Model Basin, Germany

Cloud cavitation - Preliminary classification of mechanisms

and observations on full scale ship propellers

267

G. Bark

Propeller flow field analysis by means of LDV phase sampling techniques

273

A. Stella, G. Guj

Università di Roma Tre

F. Di Felice

INSEAN, Roma, Italy

M. Elefante, F. Matera

CEIMM, Roma, Italy

A new algorithm fór numerical investigation of unsteady cavitating screw propeller

with use of variational approach

279

A.S. Achkinadze, G.M. Fridman

St-Petersburg State Marine Technical University, Russia

A method to calculate characteristics of supercavitating propeller

making use of 2-D theory for cavitating wing

285

Ando, S. Maita, K. Nakatake

Kyushu University, Ja pan

Prediction of screw cavitátion characteristics on the basis of model tests results

291

'V.P. ilyin, A.A. Roussetsky, AV. Chalov

Kry!ov Shipbuilding Research Institute, St-Petersburg, Russia

Prediction of cavitation noise radiated from a marine screw propeller

295

H. Kamiirisa

Akishima Laboratories, Mitsui Zosen Inc., Tokyo, Ja pan

New blade section design and its application

301

J. Dang, D.H. Tang, J.D. Cheng,X.X. Peng, Q. Qin, S.S. Pan

China Ship Scientific Research Center

Design of supercavitating propellers for a racing boat

309

T. Kudo, Y. Ukon

The Ship Research Institute, Tokyo, ¡a pan

FLUID MACHINERY

Turbine design for improved cavitation performance

317

Brekke

Norwegian University of Science and Technology, Trondheim, Norway

Study of hydrodynamic cavitation, in inducer centrifugal pumps

323

V.V. Pilipenko, Y.A. Semenov, O.V. Pilipenko

Institute of Technical Mechanics of the NA'S of Ukraine

Experimental investigations concerning the influence of liquid property

and speed of rotation on the inception of blade cavitation in a centrifugal pump

329

Awad, G. Ludwig, B. Stoffe!

3D cavity shape in an inducer:

experimental investigations

and numerical predictions

335

F. Joussellin, T. Maître

Laboratoire des Ecoulements Géophysiques et industriels, Grenoble, France

P. Morel

Société Européenne de Propulsioñ, Vernon, France

Cavitation at low flow rates in

centrifugal pumps

341

T. Okamura, Y. tie yama, T. Satoh

Tsuchiura Works, Hitachi Ltd. Ibasraki, Ja pan

Linear analyses of cavitation instabilities

347

Watanabe, Y. Tsujimoto

Osaka University, Japan

J.P. Franc, J.M. Michel

Laboratoire des Ecoulements Géophysiques et Industriels, Grenoble, France

A simple prediction method

of pump cávitation performance

353

Tsugawa

Kobe, Ja pan

Would turbine uprating be

allowable with respect to cavitation?

An example of vibroacoustic

diagnosis

359

B. Bajic

Brodarski Institute, Zagreb, Croatia

Development of long-lived type inducer for multi-stage pump

363

M. Maeda, M. Tagawa

International Symposium on Cavitation,

Grenoble, April 1998.

Re-Entrant Jet Modelling of Partial Cavity Flow on

Two-Dimensional Hydrofoils

Jie Dang

Maritime Research Institute,Netherlands P.O.Box28, 6700 AA Wagen ingen, The Netherlands

Telephone: +31-317-493223 Fax: +31-317-493245 Email: j.dang@marin.nI

Gert Kuiper

Maritime Research Institute, Netherlands and Delfì University of Technology P.O.Box28, 6700 AA Wageningen, The Netherlands

Telephone: +31-317-493273 Fax: +31-317-493245

Email: g.kuipermarin.ni

Abstract

A potential based panel method is developed to predict the partial cavity flow on two-dimensional

hydrofoil sections The Dirichiet type dynamic boundary condition on the cavity surface and the

Neumann type kinematic boundary condition on the wetted section surface are enforced A re entrant Jet cavity termination model is introduced A validation is accomplished by comparing the present calculations with cavitation experiments of a modified Joukowsky foil and a NACA 66(MOD) a=O 8

Re-Entrant Jet Modelling of Partial Cavity Flow

on

Two-Dimensional Hydrofoils

Jie Dang

Maritime Research Institute, Netherlands P.O.Box28, 6700 AA Wageningen, The Netherlands

Telephone: +31-317-493223 Fax: +31-317-493245 Email: j.dang@marin.nl

Abstract

A potential based panel method is developed to predict the

partial cavity flow on two-dimensional hydrofoil sections. The

Dirichiet type dynamic boundary condition on the cavity

surface and the Neumann type kinematic boundary condition on the wetted section surface are enforced. A re-entrant jet

cavity termination model is introduced. A validation is

accomplished by comparing the present calculations with

cavitation experiments

of

a modWed Joukowskyfoil and a

NACA 66(MOD) a=O.8 section.

Introduction

Cavitation on ship propeller blades is a major source of

noise and vibrations. In order to prevent excessive noise and vibrations cavitation has to be eliminated entirely orhas to be controlled in its behaviour. Since for reasons of efficiency the

propeller always operates in the wake of the hull, which is

highly non-uniform, cavitation can generally not be avoided. The increasmg speed and power installed in ships nowadays make the problems worse. It is therefore important to control the dynamic behaviour of cavitation.

Cavitation on a ship propeller is three-dimensional and

unsteady. The inflow variations into the propeller have a low

frequency, however Although the bläde loading is highly

unsteady and has to be calculated as such, it seams acceptable as a first approximation to consider the cavity as quasi-steady.

Only few authors have investigated the three-dimensional

characteristics of cavity flow (eg in [1,2]). Most attention has been given to two-dimensional cavity flows. Excellent reviews of the research on cavity flows have been given by Wu131 and Uhlman141. The classic linear solutión of cavity flow around a hydrofoil by Tulin5, Wu161 and Geurst'71 was modified by

Kinnas18' to predict the leading edge partial cavity flow.

Different from traditional linear method, this method predicts the cavity length to decrease with an increase of foil thickness.

A systematic investigation from 2 D foil to 3 D propeller

cavitation has been performed recently at MIT121191°1. A so-called split panel technique is used to avoid re-panellmg of the cavity-foil surface At the end of the cavity, a simple algebraic

expression of the pressure recovery is empirically enforced

over a given range of the cavity length at the end of the cavity

in the method But this approach of artificial recovery may

influence the final results. Also the detailed flow structure at

the trailing edge of the cavity is ignored and no insight is

Gert Kuiper

Maritime Research Institute, Netherlànds and Deift University of Technology P.O.Box28, 6700 AA Wageningen, The Netherlands

Telephone: +31-317-493273 Fax: ±31-317-493245

Email: g.kuiperniarin.nl

noise by its implosion. The re-entrant jet also influences the volume of the cavity. In this paper a method is developed to

numerically predict the re-entrant jet at the end of a

two-dimensional cavity and to assess its effect on the cavity shape

and volume.

In our investigation, a potential based lower order panel

method is used. A re-entrant jet cross section.is introduced as a

boundary of the problem, on which surface a normal velocity

into the cavity is prescribed which equals the free-stream velocity on the cavity surface. A Dirichlet type dynamic

boundary condition is enforced on the cavity surface, while

Neumann boundary conditions are enforced both on the jet

boundary and on the wetted surface of the foil

section

respectively. An initial cavity surface is firstly assumed and then iterated. The kinematic boundary condition on the cavity surface is satisfied by iteratiñg the cavity length and:shape. Upon convergence, both the dynamic boundary condition and

the kinematic boundary condition on the cavity surface are

satisfied and a re-entrant jet with a certain thickness

is

automatically formed.

Mathematical Formulation

Expression of the problem

Consider a cavity flow around a two-dimensional foil

section in an unbounded fluid as show in Figure 1. The inflow V0 has an angle of incidence a to the nose-tail line of the foil section. A steady partial cavity is formed on the surface of the

foil when the ambient pressure is low enough so that the pressure on some part of the foil surface is lower than the

cavitation inception pressure, which can be taken as the vapor pressure. A re-entrant jet is formed at the end of the cavity and flows inward to the cavity with the same speed as the speed on the cavity surface. Let us suppose that the re-entrant jet flows through an intersection surface of the jet (S in figure 1) and disappears so. Then the flow around the cavity-foil system can be treated as an inviscid fluid flow field with a total potential '»which satisfies the Laplace equation,

V2(1)=0,. (I)

If we define a perturbation potential to the inflow potential

at infinity (D0 as q, then, where,

si

f Sr

Figure 1. Partially cavitating hydrofoil with a re-entrant jet The universal solution of_{equation (4) can be obtained by}

Green's identity on a

closed boundary.

Since the two

dimensional flow field around the foil section is not a simply connected zone, a cut has to be introduced to connect the foil trailing edge to infinity. A boundary condition should also be enforced on this cut. Then we have,

2,r = fço -- In

r

ds -

f_!in

r

ds (5)

s

where,S=Sh+Sc±S;v+Sj,q is a point on the boundary S and p ¡s a point in the field

Instead of_{developing a singularity distribution code based}

on the induced velocity, we developed a potential based panel code subject to a mixed boundary condition ofDirichlet and

Neumann. When equation (5) is applied on the Neumann boundary conditions, a Fredholm integraI equationof the

second kind is obtained. But when equation (5) isused onto

the Dirichlet boundary condition, we obtain a Fredholm

integral equationofthe first kind.

Boundary conditions

The kinematic boundary condition on the surface ofthe

cavity and the foil are as follows,

ñV=O

onSbandSC (6)

Substituting equation

(2) and (3)

into (6), a boundary

condition for the perturbation potential is found,

ûn_.n. o

on Sb and S (7)

where, ñ denotes the outward normal unit vector both on the foil surface and on the cavity surface.

A dynamic boundary condition is also needed on the cavity surface which states the pressure on the cavity surface to be

constant for the steady cavity flow and equal to the vapor

pressure Pc

(8)

While from the Bernoulli's equation, we have,

Po

I-.

P'o

-

pc

+plÇ2

I

(9)

where, V denotes the constant velocity on the cavity surface

and po denotes the pressure at 'infinity. Define a cavitation

numberas,

_PoPC

°)/pv2

then, ₌

In order to prescribe easily this velocity on the cavity

surface, an alternative expression is used. Consider a curved coordinate s

along the cavity surface

starting from the

detachment point o of

the cavity near the leading edge

towards the end of the cavity. Equation (Il) can be rewritten

as;

(12)

where, Ï denotes the tangential unit vector on the cavity

surface alongs. Then,

ôs

=ï+V0V1+a

onS.

(13)

Integrating from the detachment point to any points on the

cavity, we get the prescribed potential on the cavity surface

based on the potential at s0,

onS (14)

where is the perturbation potential at the detachment point.

On the jet cross section S, only the kinematic boundary

condition is needed, which is,

(15)

where, ñ is the normal unit vector on the jet boundary toward the fluid field For the perturbation potential, we have,

1jT47.n

onS3 (16)

On the wake surface Sfl,, the velocity is considered to be continuous while the potential has a jump across the wake, but

is kept constant along the wake from the trailing edge to

infinity. It is:expressed in the perturbation potential as,

ûn - ûn

on S,, (17)

L'=Ç: q

onS,, (18)

where, A is the potential jump across the wake surface and

superscripts + and - denote the variables on the upper and

lower surfaceof_{the wake, respectively.}

A Kuna condition is enforced at the trailing edge to keep the velocity at the trailing edge finite.

At the detachment point

of the cavity, a so-called

'Brillouin-Villat condition is always demanded in potential

flow, which states that the curvature of the cavity surface at the detachment point must be continuous with the curvatureof

the foil surface at the same position. In real flow, it has been found e. g. by Shen & Peterson'1 that the detachment point is always a little bit downstream ofthe negative pressure peakof

the wetted flow or starts from the separation point of the laminar boundary layer near the leading edge. lt is still

difficult, up 'to present dày, to treat the detachment point

precisely in numerical simulation. For simplicity, the

detachment positionof_{the cavity is treated as an input for this}

method.

Substitute equation (17)and (18) into equation (5) we get,

7rÇO =

J

ço-_lnrpqds

f

--lnr,,ds

o

Sb+S+SJ S+S,+Sj +

fitçof-lnrpqds

₍₁₉₎ s

Then equation (19) can be solved under the mixed

boundary conditiOnsof(7), (14) and (16) for different parts of

the boundary together with a Kutta condition at the trailing

edge. The uiiknown potential or the normal derivative of the potential on the boundary is obtained.

Instead of solving the equations by giving a fixed cavity length and calculating the corresponding cavitation number, a

more direct way is used in the present method, which

prescribes a cavitation number but solves the equation by

iterating the cavity surface until it is converged.

Iteration scheme for the cavity surface

An estimated cavity length and shape, and a re-entrant jet cross section boundary are first assumed at 'the beginning of

the calculation. The dynamic Dirichlet boundary condition

(14) is imposed on the cavity surface, and the kinematic

(10)

on S. (Il')

and,

CL_ (28)

pVcC

D ₍₂₉₎

pV02C

are lifting coefficient and drag coefficient respectively. Here, C is the chord length of the foil.

The cavity volume coefficient is.defmed as,

c, = V/c2

(30)

Numerical Implementation

Discrete expression of the problem

Instead of using the higher order panel method, we used the lower order panel method with constant source and dipole

distributions on every panel, because the solution always approaches the analytic one when the number of panels approaches infinity. For a two-dimensional problem, CPU

time consuming is not a considerable problem.

The foil, cavity and jet surfaces are represented by N

straight line panels. The control points are located at the center

of each panel. The constant source and dipole strength

distributed on the1th

_{panel are () and}

,. respectively. Thus equation (1.9) can be expressed by the following discrete form

for every control. point i,

=A11qi1

B11()1 +C1Aço

1=1 j=l

¡=1,2,3...N (31)

where Nis the total number of panels on the boundary andA11,

B1 and C, are induction coefficients of thé dipoles and the sources on the surface and the induction coefficienti of the.

wakedipoles respectively, A. = i

lnr. .

ds 1,1 _J r

ê

₍₃₂₎ (Sb +S +S).

B,1 =

fIn r,1.ds (33) (Sb +S+Si)-and,

C,=J-Slnrjds

(34)

where (Sb+S*SJ)Jis the boundary of thejth panel on 5b,S and

Si.

The kinematic boundary conditions of equations (7) and (16) prescribe the source strengthon the wetted part of the foil

and on .the jet boundary respectively, while on the cavity surface the dynamic boundary condition of equation (14)

prescribes the dipole strength on its surface The unkÌowñs are

the source strength on the cavity surface and the dipole

strength on the foil surface.

There are different approaches. to the implementátion of the. Kutta condition 23I for steady and unsteady flows. In the

present work only the simplest numerical Kutta condition

given by Morinot'41 is chosen For two dimensional cases, this condition states that the wake dipole strength should be equal

to the difference of the dipoles. on the upper surface and on the lower surface at the trailing edge. It is written as,

Aç =

_-

(35)

th

jet. Only the kinematic boundary condition on the cavity surface is not satisfied at this moment. The potential on the

wetted part of the foil and on the cross section of the jet, and. the normal derivative of the potential on the cavity surface are calculated by solving equation (19). This calculated normal derivatives of the potential on the cavity surface are usually

not equal to the value prescribed

in equation (7.). The

difference AV,,

AV_19

ôn

ôq

¿9ço (20) req. cal.

where, subscriptreq. and caL denotes the value prescribed by equation (7).and the calcu ated value respectively.

Our aim is to re-align the cavity surface to make the flow on the cavity surface tangential to its surface.

d Ai7 AP', AV,

(21)

ds V _v0-.Ji

where, ,j is the direction orthogonal to. the cavity surface s.

Then,

s

AV

A ì

_-

_J

_j---

ds (22)

where Ai = O when s = s.

The jet thickness of S is the distance between the end

point of the cavity and the foil surface When the end point of the cavity moves to a new location after an iteration, S is also moved to this location.

Cavity volume and hydrodynamic forces

The most important parameter of the cavity is the volume. of the cavity. The accelerations of the cavity volume are the source of pressure fluctuations on the ship's afterbody. If the cavity surface is described by y=77(x), and the foil surface is described by y=J(x), then the cavity volume is obtained by the following integration,

V

=

J[ii(x)

- f(x)Ix

(23)

Xd

where, x,, denotes the position of the cavity detachment point and x,, denotes the position of the maximum cavity length (see Figure :1). The volume occupied by the jet is not considered in the calculation since it is verythin.

The lift L and drag D on the foil section is calculated by integrating the pressure all over the surface. We have,

=

_-

_{J pn, dl}

(24a)

F=_Jpndl

(24b)

and,

L=.F cosa-1 sina

(25)

D=Fsina+Fcosa

(26)

where, n and ny are the two components of the. outward

normal unit vector on the foil surface and ¡ is the arc length

along the foil surface, respectively. The pressure, under the cavity but on the foil surface between the detachment point

and the .posit ion of the jet cross section, is set toPc

a

priori in

this force calculation.

The following non-dimensional coefficients are also

Øfl M

ybpJl

tr Qt45 pf,S3fÇ1 4tJ1

panel just ahead of the detachment point of the cavity as

described in equation (14) by . Suppose the cavity starts at

panel N, and ends at panel Ne, the integration of equation (14)

is written numerically as a sum giving by the following

expression, = k Na Ne where, AS'k =(1k ''k_Ê) (37)

and 'k is the length of the kth panel.

Because of the mixed type boundary conditions, the linear

equations (31) are composed of both Fredholm equations of the second kind and the Fredhoim equations of the first kind. The matrix is no longer a diagonal dominated. Further more,

equation (36) makes the condition of the matrix even worse. A more accurate direct solver ¡s needed.

The kinematic boundary condition on the cavity surface is

used to iterate the cavity length and the cavity surface shape. The numerical form of the equation (13) is as follows,

AV,,

k=Nd...Ne (38)

where, ¿tSk is given by equation (37). The surface correction is made by changing the coordinates of the control points by z1i,

in the normal unit vector direction for every panels on the

cavity surface. The end points of panels are interpolated from

the coordinates

of the

control points, except for the

detachment point, which is actually given as input, and the iñtersection point of the cavity and jet which will be described

in the next paragraph. This numerical scheme can prevent the cavity surface from becoming saw-tooth shaped.

lt has been found in reference [I] and [15] that the re-entrant jet is hardly formed before the cavity is fully developed. lt means that the re-entrant jet will not form when the cavity is still growing. Our calculation shows the same phenomenon. At the beginning of each calculation, after the length of the cavity is assumed, a re-entrant jet surface with very small height is firstly erected vertically to the surface at this point and the cavity thkkness distribution is assumed to

increase from the detachment point to the jet linearly. If the cavity length is under estimated, the cavity will grow up very quickly over the jet boundary. 1f the end of the cavity surface goes into the foil surface, the program simply truncates that part. But if the end of the cavity is above the foil surface, the jet boundary is set there to connect this end point vertically to the foil surface. No more restrictions are enforced and the

re-entrant jet can evolve automatically. Surface panelling

The surface of the cavity and foil system is divided into

small panels Around the leading and trailing edge and around

the reentrant jet of the cavity, the cùrvature of the surface is quite large. Especially at the end of the cavity and at the thin

re-entrant jet, very fine grids are needed to obtain a converged result. The following panelling is adopted.

On the lower surface of the foil, a cosine distribution of the panel from the leading edge to trailing edge is used,

(il)

x. = 05 OScos

ir

¡=1,2,3...N1+1 (39) where, N, is the total panels number on the lower surface of the foil.

the panels on the cavity surface are used for the last 5% of the arc length on the cavity surface.

.s; =0.95(0.5-05cos

2ir)

cae

s, = 0.95+ 0.05

_{N N /2}

iN0J2...Nc+1(41)

cae cae

where, is the total panel numbers on the cavity surface.

(36) On the foil surface downstream of the cavity, the surface is

also divided into two parts,,

NACA1 6-006

Re-entrant Jet ModeF o=0.8751 3 re-entrant jet cavity surface

t-. t

t t foil surface 0.56 0.57 0.58 ¡=1,2...Ncr,/2 (40) 0.59 0.60 0.61 x/C

Figure 2. Surface paneling around the re-entrant jet surface

and the cavity end. (NACAI6-006 section at anangel of attack 40, and cavitation number 0.875 13)

Results and Discussion

Convergence test

It is certain that when the number of panels increases to

infinity, the solution converges to the correct solution

described by the analytical equation of the problem. In order

to make sure how many panels are the minimum requirements to reach a reasonable result,, the convergent test of the method was done. We took NACAI6-006 at an angle of attack 4° and

a cavitation number of 0.87513 as a test case. Figure 3 and

Figure 4 show the test results.

It is obvious that the solution is quickly converged with the increase of the panels. When both of the number of panels on

the cavity surface and on the foil surface downstream of the cavity is increased to 80, the cavity length and the re-entrant

jet thickness are converged. So, 100 panels on the cavity

surface and 100 panels on the foil surface downstream of the

cavity are recommended. A total of 260 panels all over the cavity-foil system are needed if the number of panels for the

jet _{%9Rsh9face, two sets of panels are used. Half of}

X, = 0.1

NdOw,, / 2

i1

¡=1,2...Nd0,,.,/2 (42)

and,x. = 0.1 + 0.9(0.5 - 0.5 cos - NdO,,,, / 2

dow,, - down

where, Nd0,,,,, is the total panel numbers on the foil wetted part

downstream of the cavity Figure 2 shows an example of the

panel arrangement around the re-entrant jet under the above scheme of panelling. 0.07 y/C 0.06 005 0.04 0.03 002

pro&ffi4F o

A006 §ttbntIàÇ aii1eÇlNfl41!

4° for two different cavitation numbers respectiely. lt is found that the convergence is slow toward' the final result.

From our experience, although the steps needed for the

iteration depend on the initial assumption of the cavity loo steps are always needed to achieve a result with a maximum

error, between the calculated velocity on the cavity surface

and the prescribed freestream cavity velocity, less than 1%. The maximum errors 'always occur at the intersection of the

cavity and the re-entrant jet. But on 'most part of the cavity

surface, the tangential velocity is already exactly equal to the prescribed velocity.

The influence of the detachment point on the final cavity

volume and cavity length has been well investigated by

Uhlman in reference [4]. Since we have not found a good way to. treat the position of the detachment point, a detachment is always set at the leading edge in all of the calculations in the following paragraphs. 1.0 - -I/c 0.9 0.8 0.4 40 50 60 70 100 110 "down

Figure 3. The cavity length varies with thepanel ñurnbers

0 .0060 h/C 0.0055 0.0050 0.0045 80 90.

Figure 5. Changes of the cavity shape with the iteration steps.

(NACA 16OO6 a=4°, YO.87513, linigja/C=O.37550, converged l/C=O.59279) 0.20 0.25 0.1 0.0 -0.1 -0.2 -0.3

00 01

02 03 04 05

06

07 08 09 10

'-'C

Figure 7. The final converged cavity shape With. re-entrant jet. (NACAI'6-006, ct=4°, cr=O.875l3) 0.5 0.0 -0.5 -1.0 y/C ÑACA16-006 0.2 o0.87513

a4°

Re-entrant Jet Model

NACA1 6-006 Re-entrant Jet Model

c=0.87512 a4° 1.0

i

'00 01

02 03 04 05

0.6

07 08 09

10

x/C

Figure' 8. Pressure distribution on the cavity-foil system

Predictionsfor'NACAI6 series sections

A calculation for the cavity flow with a re-entrant jet on the NACA 16-006 sectionatan angle ofattack of4° is carried out at a cavitationnumber of 0.87513. This cavitation nUmber is chosen, because the authors wanted 'to makea comparison

with the result of Uhlman's cavity termination wall model. The converged shape of the cavity and its re-entrant jet is

shown in Figure 7 and the fmal pressure distribution on the cavity and on the wetted part of the foil is drawn in Figure 8. Detailsabout the re-entrant jet are shown in Figure 2 with the

panel arrangement.

Figure 8 shows that the velocity on the cavity surface

equals to the prescribed freestream velocity. lt means that both the dynamic boundary condition and the kinematic boundary condition on the cavity surface are very well satisfiéd. Only at the control point of the last panel on the cavity, the velocity is a little bit higher than the freestream velocity, but the relative error is still less"than 1%. A stagnation point is clearly shown downstream the end of the cavity.

The flexibility of the present program makes it very easy to change the kinematic boundary condition on the jet section from equation (16) to a non-penetrative condition similar to equation (7) and find the solution with a cavity términation wall' model. Figures 9 and 10 show the cavity shape and'the pressure coefficient distribution on the cavity-foil surface with

this cavity model respectively. A stagnation' point is also

shown at the corner of the wall:. The difference of the

predicted cavity shapes by these two cavity models is shown

more clearly in Figure 11 and' Figure 12 for two cavitation 0.0040

40 50 60 70 80 90 100 110

Figure 4. The jet thickness varies with the.panel numbers

1.5

-cp

0.7

0.6

u.., y/C 0.2 y/C y/C 0.1 0.0 -0.1 -0.2 -0.3

00 01

02 03 04 05 06 07 08 09 10

ilC

Figure 9. The finally converged cavity shape with termination

wall model (F ACAI6-006, czr=4°, c=0875 13)

1.5 -cp 1.0 0.5 0.0 -0.5 -1.0 0.3 0.2 0.1 0.0 -0.1 -0.2 -03 0.3 0.2 I 0.0 -0.3

NACA16-006 Terrnination'Wall Model a=0!87513

NA A16.006 -mina

c0.E

a4°

7512

ion Wall Mdel

00 01

02 03 04 05 06

0.7

08 09 10

.rJC

Figure 10. Pressure distribution on the cavity-foil system

NACA1 6-006 i Wal Model Re-entrant J

leni

inatio t Model

a4°

00 01

02 03 04 05 06 07 08 09 10

xJC

Figure 11. Comparison of the cavity shapes for the different cavity models.

NACA1 6-006

Term nationWall Re-e

MOdl\>/

itrantJetM

a=08

odel

7513

calculations for NACA 16 series sections with different

thickness to chord ratio to study the relationships between the cavity volume, the cavity length the cavitation number and the re-entrant jet thickness. All of the results are listed in Table I

to Table 3.

Table 1. Calculated Results for NACAI 6006'ata=4°

Table 2. Calculated Results for NACA16-009 at ct=4°

Table 3. Calculated Results for NACA 16-012 at ct=4°

The special feature of the present method is the re-entrant

jet calculatiOn. Figure 13 shows the re-entrántjet thickness

versus cavitation number for three sections. It can be found in

this figure that the re-entrant jet thickness decreases with increasing cavitation number. And also ound is that the

re-entrant jet is thicker on a thinner profile (tJC=6%) and thinner on athicker profile (t/C=12%).

But when we plotted the ratio of the re-entrant jet to the maximum cavity thickness against the cavity length, we got Figure 14. It shows that the re-entrant jet thickness is always a

certain percentage of the maximùm cavity thickness

_m,

irrespective of the cavitation number and the profile thickness. This percentage is around 8% to 10%. It is also found that the maximum cavity thickness is also always located at 60% of the total cavity length (Figure 15 andX,max/Cin Table 1 to 3),

irrespective of the cavitation number and the profile thickness as well. These characteristics of the re-entrant jet could also be

found in Gilbarg's calculation of the re-entrant jet after an

obstacle61 0006 h/C 0.005 0.004 0.001 O N°CA16-0C6 -O-- N°CA16-009 N \CAI 6-012

a40

00 01

02 03 04 05 06 07 08 09

10 0.000 xJC _{06 08} 10 1.2 14

1618

Figure 12. Comparison of the cavity shapes for the different

cavity models. Figure 3. Cavity re-entrant je.t thickness WC versus cavitation

_nune?&am

analysis is

performed by a

series of

i/C

i/C xJC

Cv h/C CL CD 0.83 0.7324 , 0.06272 0.4452 0.03496 0.00549 0.6252 0.0270 0.85 0.6624 0.05737 0.3897 0,02888 0.00448 0.5920 0.0241 0.87513 0.5928 0.05197 0.3464 0.02338 0.00424 0.5614 0.0220 0.90 0.5355 0.04767 0.3124 0.01936 0.00384 0.5383 0.0206 lOO 0.3917 0.03721 0.2319 0.01106 0.00300 0.4940 0.0179 1.10 0.3066 0.03082 0.1866 0.00717 0.00275 0.4752 0.0164 1.20 0.2460 0.02607 0.1487 0.00487 0.00234 0.4635 0.0153 1.30 0.2070 0.02318 0.1220 ' 0.00365 0.00225 0.4562 0.0142 1.40 0.1724 '0.0201i 0.1014 ' 0.00264 0.00207 0.4498 0.0133 iSO 0. 1454 0.01757 0.0853 ' 0.00195 0.00185 0.4445 0.0125 1.60 0.1241 0.01549 0.0727 0.00147 0.00165' 0.4412 0.0118 1.70 0.1013 0.01378 0.0628 0.00113 0.00154 ' 0.4384 0.0112 . VC

tJC xJC

Cv WC C1 C0 0.81 ' 0.6998 . 0.04799 0.4212 0.02516 0.00339 0.5581 0.0160 0.90 ' 0.4748 0.03407 0.2841 0.01202 O.00244 0.4957 0.0117 LOO 0.3376 002593 . 0.2088 000650 0.00180 0.4622 0.0094 1.10 0.2629 0.02117 0.1613 0.00412 0.00173 0.4606 0.0087 120 0.2019 0.01683 0.1234 0.00251 0.00136 0.4522 0.0076 1.30 0.1579 0.01367 ' 0.0970 0.00159 0.00118 0,4368 0.0057 1.40 0.1271 0.01122 0.0778 ' 000105 0.00103 0.4361 0.0050 1.50 0.1029 ' 0.00915 0.0640 0.00069' 0.00088 0.4349 0.0044 VC

iJC x/C

Cv WC C'L 'C0 0.80 06544 0.03373 0,3982 0.01606 0.00226 05037 0.0076 0.90 0.4261' 0.02348 0.2608 0.00719 000148 04714 0.0051 1.00 0.2843 0.01595 0.1823 0.00321 0.00100' 0.4575 0.0040 '1.10 0.1821 0.00994 0.1195 0.00125 0.00082 0.4260 0.0015 i20 0.1258 0.00652 0.0082 0.00054 0.00048 04260 0.0008 0.003 -0.1 0.002 -0.2

1.0

x,/I

0.8 0.6 0.4 0.2 0.0 O NA A 16-006 -n- NAiA16-009

FigUre 14. Thç re-entrantjet thickness as apercentage of the cavity maximum thickness versus the cavity length

a4° _{-o- NACA16-oo9}O NACA16-006

' NÄCAI6-012

00 02 04 06 08 I/C 10

Figure 15. The maximum cavity thickness position

Comparison With other linear and nonlinear results

In order to find out the difference of the prediction results between the present method and those of other linear methods

with thickness correction and the nonlinear methods with

termination wall model, comparisons are made for NACA 16 series sections at an angle of attack 4° for different cavitation

numbers.

Uhlmant41's method is

based on velocity. A cavity

termination wall model is employed in his method. The cavity length is prescribed, while a cavitation number is calculated.

When the shape of the cavity is converged, the cavitation number is obtained. By assuming the velocity on the jet

boundary to be zero in the present method, we can also obtain the solution with a termination wall model. A comparison is shown in Table 4. It can be said that the results are very close to each other even if some differences exist for some valùes. The cavity length predicted by present method is 2% longer than the length predicted by Uhlman.

Table 4. Comparison of different approaches for the.same cavity termination wall model

correction gives very good results comparing to the present

nonlinear method for NACAI6-006 but over-predicts the

cavity length for NACAI6-Ol2. The present method predicts a cavity length decrease with increasing of thickness while the linear method predicts a cavity length increase with increasing foil thickness. Since the tendency is different, we can not say

the linear method is accurate enough for a cavity flow on a

very thin section, especially when the cavity is long.

As' for the cavity volume, a comparison is shown in Figure 18 of the present method and the linear method. The tendency that the cavity volume decreases' with increasing foil thickness

is the same but the results are very different. The linear

method seams to

over-predict the cavity volume. For

NACAI6-O'12 section, the volume predicted by linear method

'is twice as large as that predicted by the present method.

09 I/c 08 0.7 0.5 0.4 03 Ò.2 0.1 0:0 0.6 -0.5 0.4 0.3 0.2 0.1 0.0 0.03

The calculated cavity lengths are now compared with the

nonlinear results of reference [4] and shown in Figuré 16. lt is 0.01

found that the results for NACAI6-009 are very close to each _0.00

other. While the cavity predicted by present method is longer

_{02 03 04 05 06 07 08}

6% _{Ref [4]} results (non-linear) g% Present Method 12%

-06%

o A _12% O.

OÄ

0

o '

2'

A' O Ref. (4] result (linear) Present Method Uc=12% t'C=O% o t/C=6ó/o D UC=9% -- t/C=6% _{N N}

.%

j

'1 A

uiiii

Uhhnan's results Present Results Relative Differences Cav.Numbercr 0.87513 0.87513 0.00% Cavity Length VC 0.5000 0.51 00 +2.00% Cavity Volume C 0.0 1670 0.01794' +7.4% Lift Coeff. CL 1 0.53562 ' 0.51705 -3.5% 003 0.04 0.05 0.06 0.07 008 0.09 0.10 a/a

Figure 16. Comparison between present method and the

non-linear method of ref. [4]

0.04 0.05 0.06 0.07 0.08 0.09 0.10

ciJa

Figure 17. Comparison between present method and the linear

method of ref. [4] 04 06 08 I/C 10 1.0 I/C09 0.8 0.7 0.07 C 0.06 0.05 0.04 0.03 0.02

of the re-entrant jet

formation by a high speed cinephotography method. From the framing speed of the film, the velocity of the re-entrant jet was measured. The measurement shows that the maximum speed

of the re-entrant jet is a little smaller than the freestream

velocity of the cavity. To investigate the influence of the

re-entrant jet velocity on the cavity volume and length, the predictions for NACA 16-006 at an angle of attack 4° are

calculated both for a prescribed re-entrant jet velocity and for a free jet. The result is shown in Figure 19.

The free jet here means that we did not use the jet

boundary and the jet is no longer cut off. So, With the iteration continuing, the jet evolves automatically. Figure 19 shows that when the free jet is developed long enough, these tWo methods give exactly the same result. It seams that the jet boundary is

not really needed, but the free jet approach takes more

iteration and the boundary condition on the jet surface could not be satisfied finally, even if these errors do not influence the final solution. However, in three-dimensional condition, it might be easier to use this free jet approach since it is not easy to erect a jet boundary on the surface of a three dimensiónal

body. 0.15 o .0.lo NACA 16-006

a1.O a4°

y

jet not cutc

jetcutoif

if

-0.15

00 01

all over the section surface. In the real flow the section is

actually de-cambered by the boundary

layer. This

de-cambering could be treated accurately by a careful calculation of the boundary layer development on the section surface. A viscous/inviscid interaction calculation1201 for a 2-D section has shown this strong effect already. Any simple comparison based on equal angle of attack or equal lifting coefficient does

not work. In the equal angle of attack comparison, the

potential method always over-estimates the cavity length and volume because the negative pressure on the suction side is

always over-predicted. But in the equal lifting coefficient comparison, the potential theory always under-predicts the

negative pressure peak.

The authors of this paper are not going to carry out the

boundary layer calculation for this comparison because the deference of the pressure distribution may result not only from the boundary layer flow but also from the blockage of the test

section1211, even if the blockage was very small. We think that

ifthe cavity is small, the boundary layer on the suction side of

the section is not influenced too much by the cavity, and a

comparison could be done on an equivalent section that has

the same pressure distribution as measured at wetted condition.

A simple empirical method like Pinkerton's1221 is used. An

arbitrary function is used to de-camber the section,

A(f/C)=(f/C)TE .(l(x/C)2)

where, where (f/C)TE is the total de-camber amount at the trailing edge

For this NACA 66(MOD). a08 section,

(f/ÇT.E.=-OOO9 gives a good correlation between the

experiments and the calculations both for pressure distribution

and for the lift, as shown in Figure 20. The de-cambered

section geometry is plotted in Figure 21, which shows that the deformation is mainly around the trailing edge

Based on this de-cambered profile, the calculations are

done for three cavitation numbers (0.84, 0.91, and 1.00) at an angle ofattack of40 The pressure distribution is shown in

Figure 20 and

the calculated cavity length and the

corresponding lifting coefficients are listed in Table 5. The agreements are quite good.

In order to estimate, roughly, how strong the viscous effect is, a test for the same section at a conditionofa=4°and a=l.0

is calculated both for the original geometry and for the

de-cambered geometry. The results are plotted in Figure 22. As expected, the predicted cavity length on the original geometry

is 2.5 times as long as that predicted on the de-cambered

section.

But this result could not be generalized because it strongly depends on the pressure distribution on the section surface. For a NACA 66 section, which has a relative flat pressure (as shown in Figure 20) on a large part of the suction surface at wetted flow condition, the cavity length and volume are very sensitive to the pressure. On the other hand, for a section like the modified Joukowsky sectiont231, a de-cambering seam not

needed because the pressure is not so flat on the suction side.

Figure 23 shows the pressure distribution calculated by the present potential theory and the measured data in experiment

cietht js JtW1bWSk

MrtMhffitw ihluMww

Table 5. Comp.ofthe cavity length and lifting coefficients

Cavitation Number Experiments Calculations IIC CL I/C GL LOO 0.20 0.645 0.223 0.619 0.91 0.36 0.670 0.363 0.652 0.84 0.60 0.699 0.6096 0.678 02 03 04 0.5 x/Chord

Figure 19. Comp. of the cavity shapes with/without jet section Comparison with experimental measurements

In order to validate the present method, we compared the calculations with Shen's experimental measurements both for a modified Joukowsky profile1 and for a NACA 66 (MOD) a=0.8 section'71. Since the leading edge sheet cavity is very sensitive to the pressure distribution on the suction side of the profile, a careful investigationofthe influenceofthe viscosity,

the boundary layer separation and the wall effect in

experiment should be done before the comparison could be

done.

The viscosity has not only strong effects on the cavity flow, but also on wetted flow81. The experimental results show that the lifting coefflcient of NACA 66(MOD) a=0.8 section from experiment is 0.531 at an angle of attack of 3°

when the Reynolds number is 3x106, while the potential theory

predicts 0.626, which is

15% larger than

experiment.

Correspondingly, the calculated negative pressure distribution

is also hiher than the measured ones. For the cavity flow,

AvellanU9J's recent experiment shows that the leading edge cavity on a NACA 009 symmetric profile at 2.5° incidence is 30% chord length when the water speed is 20 mIs, but 45%

chord length when the water speed is 35 m/s.

This may be resulted by the differences of the boundary development on the suction side and on the pressure side at

different Reynolds numbers. For the NACA 66 (MOD) a=0.8

section, the transition points of the boundary layer on both

sides are reported to be at 13% of the chord length on the

suction side and at 89% of_{the chord length on the pressure}

0.05

0.00

-0.05

f10

without viscous correction. ihe calculated cavity lengths for

two different angle of attacks are, listed in Table 6 The

agreement is very satisfactory.

Since there is no available good experimental results for the cavity volume, no comparison is done at present.

Table 6. Comp. of cavity length calculations with experiments

u u cc Expernents Calculations 0.25 o 9 a) 15 resent method cT1 .0 a0.84 E1.0 a) C) 8

p.5

0.5 --0.2 3.8 4.3

A°'

A wetted flow

Figure 21 The.geometry of thede-cambered section.

E 03

0.249

experiment

NACA 66(MOD) t/C=9%

NACA a=0.8 fIC=2%

-1.0 i I i

--0 --0 0.2 0.4 0.6 0.8 1.0

i/Chord

Figure 20. Comparison of the pressure distributions at an angle

ofattack of40 2 0.4 o

o

_>.

0.2 0.1 -0.0 0.1 -NACA66(MOD)t/C=9%

NACA a0.8 fIC2%

2.6 1.0 0.5 0.0 Experimenl o Present Method 0.39 0.395

-Cp _{ModifiedJoukowsky Foil ai3.25°}

00

02 04 06 08 10

x/Chord

Figure 23. Comparison of pressure

disfribution

on the

modified Joukowsky profilé.

Conclusions

A potential based surface panel method is developed to

predict the cavity flow around an arbitrary two-dimensional

foil section. Lots of numerical calculations show that this

method is quite stable for different geometry. Although the convergence is still not very fast, it always converged to the

same result for different initial cavity length.

The result obtained by using a cavity termination wall

model in present method shows a good agreement with

Uhhnan's non-linear termination wall result. Lots of calculations for NACA 16 series sections with three different ,

thickness to chord ratio (6%, 9% and: 12%) have been

performed. The predicted cavity lengths and volumes with re-entrant jet modeliñg have been compared with the Uhiman's nonlinear results and the linear results with thickness

corrections respectively The present calculations show the

same trend like other non linear methods that the cavity

Iengthdecreases with increäÑing foil thickness.

As for the cavity volUme, the predicted results for NACA 16-012 are only half of the cavity völumes predicted by the

linear méthod. This valué is also smaller than Uhlman's

nonlinearresults. The cavity lengthspredicted with the present re-entrant jet model are larger than those of the termination wall model for NACA 16-006, but shorter than the results for

NACA 16-012. The re-entrant jet thickness also decreases

with increasing foil thickness

It is very iniportant to have found that, at least in our calculation cases, the re-entrant jet thickness is always a

certain percentage

of the

cavity maximum thickness,

irrespective of the cavitation number, the cavity length and the foil thickness The locations of the maximum cavity thickness

for different conditions are also fixed at 60% of the cavity

length from thedetachment point.

The present method is validated by comparing with Shen's

experimental data.. Since the experiments show a strong

viscous effects both for the cavity flow and the wetted flow around NACA 66 MOD) a=08section the comparisons were

done by de-cambering the original NACA section into a

00 0.6 0.8 1.0

IChord

..sults and the

,-nother comparison, is made for a modilied Joukowsky

profile at two different angles of atiack. Because the pressure

distribution on the suction side of the section is not very flat,

the calculated cavity length is not so sensitive to the pressure

difference between the calculation and the measurement. A

good agreement has been found between the present

calculation and the experimental data. Viscous correction

seams not needed for this case.

In summary, the re-entrant jet modéling with the potential

theory for 2-dimensional cavity flows is a quite stable and

convergent method. If the viscous effect could be included in

the method, it could provide a rather precise prediction. A benchmark test of the cavity volume for steady condition

should be carried out to verify the present theory. A further extension of the method to three-dimensional predictions and unsteady cavity flow around hydrofoils and propeller blades

seams feasible.

Acknowledgments

This work was supported by the Maritime Research Institute Netherlands (MARIN). The authors would like to

express their appreciation to the comments of H. C. J. van Wijngaarden and to the discussions with Prof. S. L.. Ceccio when he was visiting MARIN.

References

[1] D. F. de Lange, 'Observation and Modeling of Cloud Formation behind a Sheet Cavity', Ph. D. Thesis of

Twente University, April, 1996.

[2] S. A. Kinnas & N. E. Fine, 'A Numerical nonlinear

Analysis of the Flow around Two- and

Three-Dimensional Partially Cavitating Hydrofoils', Journal of

Fluid Mechanics, Vol. 254, 1993, pp. 151-.181..

T. Y. Wu, 'Cavity and Wake Flow', Annual Review of

Flùid Mechanics', Vol. 4, 1972.

[4] J.S. Uhlman ir, 'The Surface Singularity Method Applied

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