Influence of ventilationon the shape of slender axisymmetric cavities

Igor Nesteruk

Institute of Hydromechanics National Academy of Sciences of Ukraine

inesteruk@yahoo.com

CAV2012, August 14-16, 2012, Singapore

The steady flow pattern and the following assumptions are used:

1. External water flow is axisymmetric potential, inviscid and incompressible

2. Internal gas flow is one dimensional, inviscid and incompressible

3. Gravity and capillarity forces are neglected4. Cavitator, cavity and hull are slender

1/ cm LR

The external water flow potential and Bernoulli equation (Cole, 1968)

)ln()(ln)(

)(ln),,(24

*2

2

OxBrxA

xAxrx

1

0

ln)()(

2

1

dxxsignd

dA 2ln)()( xAxB

r

r*dx

dFF)x(A ,

)x(R

)x(F,

)(ln1)( 22

2222

)( O

dx

Rd

dr

d

dx

dxC

xRrp

Bernoulli and continuity equations for the internal gas flow

2

)(

2)(

220

0

xVVpxp gg

QxRxRxVRRV bb )()()()( 2220

200

222220

20

22

2

0 )(

1

)(

1)(

bb

gp RRRRU

Qxc

Basic differential equation and initial conditions (Manova, Nesteruk&Shepetyuk 2011)

])(

1[

ln 2220

2

22

bRRa

dx

Rd

ln240

2

2

UR

Qg 220

20 ]1[

R

Ra b

20

0

)(2

U

pp

At the cavity surface:

(1)

; ;

Initial conditions at : ;

flowgasinpwaterinp xCxC )()(

0x 1R dxdR /

Cavities on cylindrical hulls ;)( 0 constRxR bb

;0 Veeff

:)1(. Eq

)/()(ln 22 UUaVe gg

Base ventilated cavity on a cylindrical hull

;/2 dxdRu

)2()1(

1

)(

1

ln

)1(

2

420

20

2

222

bb

eff

RRR

Ru

Cavities on cylindrical hulls, (Manova, Nesteruk&Shepetyuk 2011)

0

;1.00

;1.0 0bb RR

Semi-length and maximum radius of ventilated cavities at:

= 0; 0.5; 0.8; 0.9; 0.99 (curves 1-5 respectively)

Cavities on cylindrical hulls,Critical values of ventilation rate, corresponding to unlimited cavities

0

;1.00 ;1.00bb RR = 0; 0.5; 0.8; 0.9 (curves 1-4)

0)1(

1

)(

1

ln

)1(2

2

0

2

0

2

2

2

bbm

meff

RRR

R

:)2(. Eq

ln/01 acr

01 crVe

:)1/()ln2( 20

200 bcr Rfor

.)(2 Figinlinessolidexistscr

Base cavities on cylindrical hulls, Ventilation diminishes the length (Nesteruk&Shepetyuk

2011)

;00 ;1.0 0bb RR Base cavity length at:= 0; 0.5; 0.8; 0.9(curv.1-

4)

0

0 1.0

1.0 1.0

Asymptotic solution at small values of the ventilation rate

)()()()( 22)2(

2)1(

2 OxRxRxR

12ln2

)(2

2)1( x

xxR eff

2

02

12

12

)1(

1

0

22

)2()()(

)(x

b

x

xRxR

dxdxxR

1.00

;1.0

))1(.,51(

99.0;9.0;8.0;5.0;00

eqcurves

RR bb

)(: )1( xRlinedashed

Cavities on conical-cylindrical hulls.Calculations at the fixed conical part length and different values of cylinder radius (Nesteruk&Shepetyuk 2012)

171 x

1R

6.11 R 7.11 R

21 R

2.21 R

CONCLUSIONS

Ventilated steady slender axysimmetric cavity is considered with the use of one-dimensional inviscid flow of the incompressible gas in the channel between the cavity surface and the body of revolution. The non-linear differential equation and its numerical and asymptotic solutions were obtained.

For the disc and cone cavitators the ventilation can sufficiently

increase the cavity dimensions and its rate is limited by two critical values. Ventilation sufficiently decreases the base cavity length.

Examples of calculations for cylindrical and cone-cylindrical shapes of the body located in the cavity are presented. It was shown that the cavity shape depends sufficiently on the values of Ve and the cavitation number at the same fixed cross-section.

Presented theoretical results allow explaining the

experimental facts of both a weak and hysteresis dependence of the cavity length on ventilation and its abrupt increase.

Acknowledgment

The author thanks Professor Kai Yan for very useful discussions and his presentation of the paper on CAV2012.