Comparison between calculated and measured results of turbulent boundary layers around ship models

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TECHNISCHE UNIVERSITEi

Laboratorium voor

Scheepshydromechanlca

Archlef

Comparison between Calculated and Measured Resulti.e.

_{m K meg 2, 2628 CD Delft}

of Turbulent Boundary Layers around Ship Models Tel.: 015-78687'3

- Far 015.781838

1. Introduction

Ship resistance in still water is classified into two compo-nents, namely wave-making resistance and viscous

resist-ance, of which viscous resistance component occupies a

large percentage of total resistance of a ship. Therefore, at the initial design stage of the ship hull form, information on viscous resistance is very important not only to predict the propulsive performance but also to improve the ship form. The nature of viscous flow field at the stern is related to self-propulsion factors, ship maneuverability and cavitation erosion of propeller blades. Recently, it was made clear that propeller excited vibratory forces are mainly caused by the

unsteady cavitation, which results from the operation of a

propeller in a non-uniform flow field at the stern.

Furthermore, it is well-known that there exists the scale effect on the viscous resistance and viscous flow field. It is

desirable that the evaluation of scale effect or the extra-polation of the model test results to the full scale can be

performed under theoretical considerations.

The three-dimensional boundary layer theory may be

considered to be a basis for the investigation into the above

problems. Recently, many studies on the boundary layer

around

ships were made, for

instance,

by Uberoi"),

Webster and Fluang(2), Hatano et al.(3), von Kerczek(4),

Himeno and Tanaka(5), Larsson(6), Okuno(7) and Gadd(8)

according to the integral method, and by Cebeci et al.(), Chang" Soejima and Yamazaki" 1) and Abdelmeguid et al." 2) according to the differential method. All of them,

except the last one are based on the conventional boundary layer assumption that the boundary layer thickness is very

thin and the pressure

in the boundary layer

is almost constant.

Some experimental investigations,

for

instance, by

Bibliotheek van de

Afdeling Scheepshouw- en _{Scheepvaartkunde}

Technische Hooeschool, Delft

DOCUMENTATIE

_I:

_{KSI 153}

DATUM'

I

I 2

NOV. 279

* Resistance & Propulsion Research Laboratory, Nagasaki Technical Institute, Technical Headquarters

Tetsuro Nagamatsu*

In order to examine the accuracy of the method of the boundary layer calculation around a ship hull and to make clear the limit of their application, comparison is made of the results of calculations and measurements of velocity distribution in the boundary

layer around ship models of different hull forms, namely, Wigley's form, a cargo ship and a tanker.

Calculations were made by not only the conventional integral method based on the assumption of thin boundary layer, but also attempts were made to take into account the effect of the longitudinal curvature on pressure variation across the boundary layer. Measurements were carried out using a five-hole spherical probe and a three-hole probe of NPL type in the towing tank. Each

ship model was 8.0 m long and its Reynolds number exceeded 107.

For the Wigley model, the calculated velocity profiles and integral parameters show a fairly good agreement with the measured

results. However, for the cargo ship and the tanker models, there are considerable differences between the measured and the

calcu-lated results in the flow field around their afterbodies. By considering the pressure variation across the boundary layer, agreement

between the calculated and the measured results is somewhat improved, but the discrepancy in the stern part is still large.

It is considered therefore that further investigations are necessary to improve the prediction of flow at the stern, based more radical approach to the nature of flow including separation.

Larsson(6), Sehsah" 3), Nagamatsu" 4)

and Hatano et

al!' 5), however, have indicated that the boundary layer

thickness increases rapidly towards stern and the pressure

in the thick boundary layer varies appreciably. For a body of revolution, a few works of treating with the problem of thick axisymmetric boundary layer were made, for

instance, by Patel and Lee(16), and Dyne" 7).

In the present study, an attempt was made to improve

the conventional method of calculation taking into account the pressure variation across the thick boundary layer at the

stern of conventional ships. The conventional integral

method of calculation was modified by adding the terms

arising from longitudinal curvature effect.

The results of calculations were then compared with

those of measurements on ship models of Wigley's form, a cargo ship and a tanker. And the accuracy and the applica-bility of the boundary layer calculation are discussed.

2. Calculation method 2.1 Coordinate system

Since ship hulls are constructed by complicated and non-developable surface, a Cartesian coordinate system is not suitable and therefore an orthogonal curvilinear coordinate system is commonly used for three-dimensional boundary

layer calculations.

The streamline coordinate system, which is one of the

orthogonal curvilinear coordinate system, is convenient for the calculation of three-dimensional boundary layer

accord-ing to the integral calculation method and for the physical

understandings of the boundary layer characteristics. One axis of the system is -axis outward normal to the body surface. The other two axes are denoted by X and

IVITB 133 July 1979

an inviscid fluid and their orthogonal trajectories, i.e. in the direction of equi-potential lines, respectively.

However, the determination of the streamline

coordi-nate, especially equi-potential lines, is not so easy.

Further-more, an equi-potential line does not coincide, in general, with a girth-line. Therefore numerical calculation of the derivatives in the direction of equi-potential lines is rather troublesome. To facilitate the numerical calculation of the boundary layer, Himeno and Tanaka(5) introduced a new

coordinate (,

0) which refers to the mathematical re-presentation of a ship hull derived by von Kerczek and

Tuck" 8 )

Cartesian coordinate system (x, y, z) , the streamline

coordinate system and this new coordinate system are shown in Fig. 1. Let us examine the relationship among

them. The origin of Cartesian coordinate system is located at midship on the still water plane and their unit vectors are

denoted by (i, j, k).

By means of conformal mapping, a cross section of the hull

is transformed to the unit circle by the mapping

function.

y + iz =

anwei(3-2n)o

(1) n=1

where i means imaginary unit.

The coefficients an(x) are expressed by the polynominal in x

an (x) 1

,77=1

Fig. 1 Coordinate systems

(2)

and is defined by the lines of 0= constant on the hull

sur-face and approximately holds

x=,

(3)

Denoting the position vector on the hull surface by

r(x, y, z), the unit vectors of 0) coordinate system are obtained from

ar

ar,

ar ,

et = _/1 1, e, - ao ao 1, e e0 x et, (4)

On the other hand, the unit vector eN in the direction of streamline is represented by use of the velocity components

(ux, uy, uz) on the body in an inviscid fluid,

ux .

ex

+ - j + k (5)

U U U

where

u = (ux2 +uy2 uz2 )1/2 (6)

a a a

+vo

N- 3 ao

and the unit vector eri (ei7x,

y, e)

in the direction of

equi-potential line is derived as

en = ex x (7)

Let hi, h2 and h3 denote the metric coefficients c

streamline coordinates. Then, according to the for

a.

a.

a 1 a i a

r= ax` + ayi + az^ = hi ax ex

h2 at h,

f the

mula

a e

an "

(8)

the derivatives with respect to streamline and equi-potential

line, which are included in the governing equation of the boundary layer, can be transformed to the derivatives of

E and 0 coordinates, i.e.

1a

a a 3

al

- ex. R4. + ex- r

ao _a, at

+0, ao

1

aa

_

,t

7A

-

-"" -

:

h30 = a2

+132 a0 where

al = ex- Fit= u , az = en- erx

ux

ay ay

az az

)+

uy ay

= ex- "= [ u

ao ao

U 30

+Uuz azao

ay ay

az az 132 e77' 17°- [erix ao ' ay az

eY

+ erz01/G

ay 2

aZ G =

_'au'

_{+ (ao}

These relations show that numerical calculation of the boundary layer can be performed on the new coordinate

system 0) after the governing equation based on the streamline coordinate system may be transformed to the

new coordinate system.

Since the streamline coordinates are an orthogonal curvi-linear coordinates, the relationships between rate of change

of the unit vectors (ex, et, en) and their metric coefficients

(h,, h 2, 113) are derived as follows" 9),

ae_

x ah 1 en ah, 1

ax

h, 3

/73 an

1

aen

exa/73_ e- ah3

an

hi

ax n2

ag-From these equations, the geodesic curvature K and

K2, and the normal curvature K3 and K4 are obt

ained. (10) = (S, ar = a ]/G (S, , (11)

11 aen

_i

3h3 K11 =ex. .h3 an h1h3, ax 1 aex 1 ahr K2 ell h1 ax

hih,

1 ae, K 3=

e--ax iae17 ii ah3 K4 -7 erh3

_{ar, =- h2h3}

K 3 and K4 are often 'called 'longitudinal curvature'

and 'transversecurvature:: respectively. 2.2 Governing equations

The steady-state Navier-Stokes equations fOr an

incom-pressible. fluid based on the orthogonal 'curvilinear co-'

ordinate., system' are written 0.1'

as

uai

w ,aw

v av

vc.A..)2)

hi 3X

hi ax hi ax ap v r a a 02 CO 2 (h3o...)3)]

p

h,ax

U au W aW V av

-- (VW

W.A.) 1),

132 4 h2

h2 ' 3' aP V a a + [ (h3w3) (V7 10-Y1)1 = p ,h

hih3 3X

u au,

w aw

v av

___ ow 2 .

h3 an h3 an h3 ap v a a =_ _{+ [}

_oicao

_[iNc,),) p,.h377

hih2

a where a La 6)1= h2 h 3[ (1330 an (h2w)] a a .

(02-

_hih3E,ari:i

hio

_ax(h30,1 1 a 3 h'ih2 ax(h 2

=

(h ,u)

and u,,,w and v are the components of velocity in the X, and n directions, respectively. The continuity equation: is

1 'al a a

[

(u23$

_{whah3)+ vh1h2)1= 0 .05)}

hih2h3

axhh

an

The conventional boundary layer approximation is de,

rived from' an assumption that the boundary layer thickness

is very small compared' with an appropriate radius of

curvature R of a body surface, i.e., 8

= R<ii

1 h

hih2

--(113) 4),

From this approximation the following. relations for

order of magnitude are derived!.

3 1

= 0()1,.w = 0(e) _! 4

and

8h II 3h 3

= 01(e)' .

-

1()

'Furthermore, it may be generally assumed that corre-sponds identically with the physical! distance from the body'

surface, vizl. 21)

h2 (181

In case of the boundary layer around a ship hull,

hciw-ever, the boundary layer thickness around after body

increases rapidly towards stern with decreasing radius of 'curvature of the body surface_ The boundary 'layer thick-ness in such a region may not be much smaller than the iradius of curvature. Therefore, in the present study', the followiing relations are introduced instead] of eq. !t1 7!)' to

take into account this effect.

and a/73

=01!',

Oft)

.4

-From these relations (1i6X and k 19), the Navier-Stokes equation (13) are reduced to the three-dimensional] bound-ary layer equation

u au

w au. v' ,au uv ah, v21 h3

hj, ax h2 a h3 an h,h,,, art h1h3 ax 1 ap

i arx

p hi ax

p h, 4

ti2 ahl ah 3 1 ap

=___

hih,

h2h3 p 172n u av

w av

av u2 ahl uv at), +-ha, 3X h2 an huh3, on h1h3 ax 1 ap arr, + (20-o)

p h3an

P

where TX Tx and 7-17 are shear stress components in X and n

direction, respectively.

If !Tx and 7%17 are represented by

11.11T13 133 Airy 1979

(116)

(11 7)

(19)

i(2013)

Where u'w' and v'w' are the so-called Reynolds stress

eq.. (20), hold's for both laminar and turbulent flow.

the conventional boundary layer equations derived from the relations of leis., (16). and] (1,7), the left hand

"3 + + 3 + 1 WW1) -1 -v2 h3 1 In (21) 1 (12) (20a)

IVITI3 133 July 1979

side of eq. (20b) is put equal to zero. This means that the pressure in the boundary layer

is constant in the

direction.

With the relation of (19), however, the left hand side

of eq. (20b) can not be put equal to zero. (

h

1 oh'

,h2

1 ah,

and ( ) are normal curvatures of the

stream-h2h3

line and equi-potential line, and are denoted by K3 and K4 respectively as described in the previous section. Assuming

that K3 and K4 are of the same order and approximately constant across the boundary layer, and considering the experimental fact that u- is much larger than v2, eq. (20b) can be reduced to

K3u2-

1 Op P

(p- pw)=- K3 f .t12CA.

0 1

--(Pe-Pw)=-K3jou-oT

Eliminating pw from eqs. (23) and (24),

The pressure gradients fore, become

1 ap 1 ape 1 a

[K3Su2dN

p h3an p h3077 h3

If the flow just outside the boundary layer is assumed to be parallel to the body surface, eqs. (20a) and (20c)

may be reduced to

Lie aue

ve aue+ueve ah,

h, ax

h,

077

h,h, an

h,h, ax

ve2 ah3

1 ape p hi ax

Li ave ve ave ue2 ah, ueve an3

_ +

h, ax

h3 an

h,h, an

h1h3 OX 1 aPe

_

p h3an (22)

Integrating eq. (22) from zero to in the direction,

the pressure p at is obtained by

where pw expresses the pressure on the body surface.

Similarly, the pressure Pe at the outer edge of the

boundary layer is obtained by

P- =P+ K3f6u2d

(25)

P P

where ue and ye are the value of u and v just outside of the boundary layer and are assumed to be equal to the

velocity on the body surface in an inviscid fluid.

Furthermore, if the streamline coordinate system is

em-ployed, since ue

is equal to U and ye

is zero, eq. (27)

can be simplified as

U aU

1 ape

ax p

hiax

u2 ah, 1 ape

h1h3 017 p h3017

Substituting eqs.

(26) and (28)

into eqs. (20a) and (20c), the governing equations of three-dimensional

bound-ary layer based on the streamline coordinate system can be written as

u au

au

v au

+w--+

- uvK2+

K,

hi ax

h, an

(23)

U au

3 _{[K3f6u2 ciT]+} 1 aTx

h, ax

h, ax

10

fou2 1

ar,

(24) =

K 2 - - - - [K 3

h, ax

S]+

where Ki, K2 and K3 are curvatures of the body surface as defined by eq. (12) and can be evaluated from the ge-ometry of ship form and the streamlines on the body

sur-face in an inviscid fluid.

there-u av

+w +

av

v av

u2 K2- uvKi

hi ax

n h 3 an

(27) 1 a

7,v

+

_{K2(022+ 011 +6- K3b-}

hi-77-(K31)-pU2 where (28) (29)

2.3 Numerical calculation method

There are two methods solving the boundary layer

equation. One is the integral method and the other the

differential method.

The method employed in the present study belongs to

the integral method.

From eq. (29) and the continuity equation, we may

obtain the momentum integral equations.

aoll1 au

_{(2e-ft 81*- 2K 31)} ao,,

ax

U h, ax non

1 a _TN.w - K11011-0221

(K3I)--h, ax

pU2 1 _ao21 202, aU _a022 + -,-

2(021

hi ax

U hi3X

h3an (30) h

ffi

LI 2 cR-

()

(31) 0 Ic U 1 a

p

1 ape a 6 [K3 u2d (26)

p n,ax

p n,ax

h, 3X

the and directions,

an

---and rx,, ---and 7,, are the wall shear stress in X ---and ri

direc-tions.

A difference between eq. (30) and the conventional

momentum integral equation is whether the terms of

curva-ture K3 (underlined in the above eq.) are included or not.

Coles' wall-wake law(22) is used as the velocity profile family.

US _{(71 cos0}

_{Uk f (w)}

v u.10( sin0+ Uk2f

where 1 rr"'

fo = 11.(t-1*)

B, f

[1

cos(6-)]

K and K = 0.41 , B =5.0 (34)

k , and k" are a kind of shape factor and are obtained

from the values of eq. (33) at = 8, viz.

u* u*.8 u.6

k1 - 1 fo(

) cosO,k2-'r--f0(

) sint3 (35)

U v U v

Eq. (35) expresses the skin friction relation. The mo-mentum integral equation (30) contains seven unknown

variables: 0, ,, 0 1 2 2 1 ° 2 2 ,61*,Txw and Tnw However, since these variables can be rewritten by use of eq. (32), the

resulting momentum integral equation contains five

un-knowns, namely 5, u*, 3, k1 and k2. Therefore the two

mo-mentum integral equations (30) and the two skin friction

relations (35)

are not sufficient to close the system of

equations. Here an additional equation, which is called an

auxiliary equation, is needed to determine the five

un-knowns. For example, an energy equation, a moment of

momentum equation and an entrainment equation are

often used as the auxiliary equation. More simply, the

shape factor H may be taken as constant.

In the present study, an entrainment equation for

three-dimensional flow is used as the auxiliary equation. This equation expresses the rate of entraining fluid outside the

boundary layer into the boundary layer and is derived from

the integration of the continuity equation in

direction.

a(s

6 t)

362 1 _au*

)-(36)

1-1,3X h,a77

F (5

61*)(Kr/-

U hi ax

Head(23) has given the relation

F =0.0306 (H._ 3.0)-0.653 ₍₃₇₎ for two-dimensional flow. This relation is applied along the

streamline for the three-dimensional flow.

Eqs. (30), (35) and (36) are basic equations to calculate

the boundary layer around ship

hulls numerically. The derivatives with respect to X and ri in these equations are transformed to the derivatives with respect to and 0 by

IVITI3 133 July 1979

use of the relation of (9) and (10).

The velocity distributions on the

hull surface in an

inviscid fluid, the geodesic and normal curvatures and the unit vectors of the streamlines are obtained from the

poten-tial flow field around the ship hull. In the present study, Hess & Smith method(24) was used for the calculations.

3. Comparison with experiments

The velocity

measurements in the boundary layer

around three different types of ship models, Wigley's

form, a cargo ship and a tanker were carried out in Naga-saki Experimental Tank, MHI(25).

Each model was 8.0 m long. Such considerably large size

was adopted for the purpose of carrying out the measure-ments with high accuracy and at high Reynolds numbers.

She was provided with a row of studs at s.s. 9-1/2 as turbu-lent stimulator. The studs were 1.2 mm in height and fitted with spacing of 10 mm between each other.

Two different probes were used for the measurements.

A three-hole probe of NPL type which was made of three steel pipes of 1 mm in diameter was used for measuring

the velocities in thinner boundary layer around forebody or near the body surface. A five-hole spherical probe of 7 mm

in diameter was used for measuring not only the velocity components in the directions of the streamlines and the

equi-potential lines but also the velocity component normal to the body surface and static pressure in thicker boundary

layer.

On the other hand, the boundary layer calculations for each model were performed by means of two methods,

namrly the method described

in the previous section

(which shall be called "present method" hereafter) and the conventional method which is almost the same as Himeno

and Tanaka's calculation method(5) except the auxiliary

equation; Himeno and Tanaka used moment of momentum

equation, while the author used entrainment equation. Reynolds numbers for the calculation coincided with the

corresponding experiments but free surface was assumed to be rigid wall.

The number of panels dividing hull surface (a quarter of

double hull) for the calculation of potential flow was 260 for the Wigley's form, 336 for the cargo ship and 345 for the tanker. For all the ship models, the starting points for the turbulent boundary layer calculations were taken at fore end which was assumed to correspond to a virtual origin of turbulent flow. The comparisons between

calcu-lated and measured results for each model are described in the following sections.

3.1 Case of Wigley's form

Wigley's form is represented by a simple parabolic

ex-pression. The body plan and principal particulars are shown

in Fig. 2. The model was towed with a forward speed of

1.50 m/s which corresponded to Reynolds number of

1.25 x 10'. The velocity measurements were made along

5

(33)

... .... .. .. .

--MTB 133 July 1979

the constant depths, namely Z = 60 mm and Z = 180

mm as shown in Fig. 2.

Fig. 3 compares the calculations of the velocity profiles in the boundary layer with the measurements. The numbers

in Fig. 3 correspond to those of the measuring position

shown in Fig. 2. The abscissa is taken as the dimensional distance normal to the body surface.

The normal velocity component w in the calculation is

Fig. 2 Body plan and measuring positions of Wigley model

Measurement

Calculation

Present method Conventional method

measured by 5-hole spherical probe

measured by 3- hole probe of NPL type

50 (mm) _ 0 50 C (mm) 10

0f

obtained from the continuity equation by use of the

calcu-lated values of u and v, which are the velocity components

in the directions of the streamline and the equi-potential

line, respectively.

From Fig. 3, it is noticed that the velocity components in streamline direction, which are commonly called "main

flow," show a good agreement between the calculations and

the measurements before the stern region except for the

measuring position

The velocity components in the direction of the

equi-potential line, which are commonly called "cross flow,"

are very small and again show a good agreement except the

measuring positions 9 and © near bow.

The calculated and the measured momentum thickness

0.1[

0

o Measurement

Calculation by the present method Calculation by the conventional method

C

5

&Station

Fig. 4 Comparison of calculated and measured boundary

layer parameters

Fig. 3 Comparison of calculated and measured velocity profiles o QO 1.0 0.5 "Um do 0 0 0 s.s. 0 50 100 150 (mm) 0 00 S.S.2 1/2 0.5 0 50 100 (mm) u/U 0 as.V2 WU, tr/U 150 200

.

I

...

I 50 100 150 (mm) 0 40 100 C (mm) 150 200 1.4 1.3 1.2 0.004 0.003 0.002 0.001 0 Z=60mm 1 5 1 1.3 1 2 0.004 0.003 0.002 0.001 Z 180mm 26n 0 (mm) 50 0 50 (..1 wit 10 05 WU, 1.0 0.5 0 S.S.8 1/2 _{(Ds.s.5 1/2} ' 8.000m L,,,,/B= 10.000 B/d= 1.600 G=0.444 L L u/Us., v/1100 w/Um u/U., AP o 5 &Station 10 WU 10 WU= 0.

0

0.5 s.s 5 0 -05 C)s.s21/2 imWdlb tr/U.0 I I I

©

-FP

and shape factor as representatives of integral parameters are in close agreement as compared in Fig. 4.

There is little difference between the results from the present calculation method and the conventional one as

shown in Figs. 3 and 4. This reason may be considered that

the Wigley's form is fine and therefore the ratio of the

boundary layer thickness to the longitudinal curvature is

very small even at the afterbody. However, it is to be noted that there is slight variation of pressure across the boundary layer as shown in Fig. 5. The pressure variation predicted

by the present method seems to be slightly less than. the

measured one but agrees qualitatively.

3.2 Case of cargo ship model

A cargo ship model used('4) is expressed by eqs. (1) and

(2) according to the given The body plan and

princi-10 u/U... 00o

Measurement

Calculation by the present method

(mm)

@ S.S.8 1/2

Fig. 5 Pressure variation in boundary layer

0-00

4/Un, @ S.S.8

pal particulars are shown in Fig. 6. The model speed was

1.50 m/s, which corresponded to Reynolds number of

1.166x 10'.

The velocity measurements were made at several posi-tions shown in Fig. 6 and the results are compared with the calculated ones in Fig. 7.

In Fig. 7, the following points are observed.

As for the main flow, agreement between the

calcu-lated and the measured results is fairly well before S.S.2,

except the positions of 0 and 0 near free surface.

After S.S.1, however, the calcu[ations do not provide good prediction, especially for the flow near separation

which was measured, for example, at the measuring

posi-tions of g and © .

The magnitude of measured cross flow is rather small

everywhere and agrees fairly well with the calculated

results. However, the so-called "reversed cross flow" was measured after S.S.1., where the flow can not be

predict-1,-8.000m L,,,,/B=6.468 B/d=2.365 C6=0.571 0 5 0.5 S.S.7 S.S7 o/U,, 50 0 50 (mm)

(.0

0 0 o 0 u/Uoc. 0 S.S.3

...

1

....

Fig. 7 Comparison of calculated and measured velocity profiles (continued on p. 8) (orn)

IVITB 133 July 1979

01 or te/Uo.

.

50 100

on,)200

112 200 0 5'0 100 10

Meas.mgP09 or

Fig. 6 Body plan and measuring positions of cargo ship

model 7 -0 1 0 0.1 0.1 0 0 1 8 S.S.21/2 0 o 00 0 1[ 2- 0 '-0.1 0.1

LU

0.1 0 (ID S.S.2 V2 0 0 0 0 0 CI) 50 100 (mm) 0 S.S.1. o 0 o 1 0 ° 50 100 (mm) ® S.S.1 0 0 50 100 (mini 150 50 100 (mm) 1 1 0 ° (f) S.S.5 n/Uo: 0.5 I u/U. o @ S5.5 50 (mm) 50 rnm 0 0 50 (mm) 000 _{o o o} Measurement Calculation ° Present method

4/U. Conventional method

4,/Un, 10 0.5 (2) 053 05 05 0 V

A.

0 (2)1 f0 0

NITB 133 July 1979 0 0 0 0 0 0 o o

00 00 000

°L'' 0 ° _u)U, () S.5.2

...

000,

.

0 50 100 lbo 0 S s2

Fig. 7 Comparison of calculated and measured velocity profiles

-

...

0 50 I0 0 200 AO Imm) rit'

...

50 191 ISO 2SO 250 Irnm) s.s .." 0 0 0 p 0 0 (....,r.,...ea,.Fr 0.010-0 0.010-00.010-05 0

--20 -3 -40 -50 90 0 010-0 010-0010-05 28h 0.010 0 005 0 26,, _ 0=0 -10_-20 -30 - 40 -50 --I 90 10 -20 -30 -40 -50 0-90 -150 -140 -:130 -120 -110 0 0 -50 -140 -30 -20 10 0 -50 -30 -20 -10 0 1.5 14 1.3 1.2 0.002 0.001 0 -90 1.6 1.5 1.4 1.3 1.2 1.5 1.3 1.2 0.002 0.001

0-90

1 6 1 5 1 4 1 2 S.S.5 0 H 16 '5 14 13 12 0.010 0 005 0 1.6 1.5 1.4 13 1.2-S.S.3 -90 o Measurement

Calculation by the present method the conventional method - - - Calculation by S.S.7 8- 0 -10 -20 -30 -40 -50 28, 41' 0 11* -60 -30 0 (deg) S52 9-0 0 -60 -30 ti (deg) SS]

-0 0 40 - -'20 -10 0 (deg) S,S.1/2 0

....

0 -1 o'E

se..

0 i 0 5 0 1 05 200 250 0 0

0-0 100 150 (n.) C)051 200 250 0 0

...

0 200 250 ₅₀ 1110 10 20 2,O 0 0 S01/2 0 so 0 001/2 0 0 0 ,U 1..., 01

.7.71r.r5----O-.

,0000, o -4,01-. 0 50 100 15CH-Aro-Th 0 50 100 150 200 250 0 50 100 150 200 Ek6 (mm) (M)

(deg) 8 (deg) 8 (deg)

Fig. 8 Comparison of calculated and measured boundary layer parameters

(no) S' (on') 0. 1 0-I ° 1 1 0 50 100 150 (mm) 0 0.5.1 05 0I 0 (m')

50 40

0 '0 0.5 o 0 o, 0 40

ed by the calculation.

(3) It is worth noticing that the measured normal veloci-ty component is in the same order of magnitude as the cross flow component. This fact may be a key to

im-prove the prediction method for flow field at stern. The girthwise distributions of momentum thickness and shape factor at each measuring section are shown in Fig. 8. In these figures 0 is taken as abscissa instead of girth length.

The value of 90° corresponds to the keel and the value of

07 the still water line.

For the momentum thickness 01, a fairly good

agree-ment can be seen between the calculated and the measured

values. The shape factor H, however, is less accurately predicted than the momentum thickness and there is a large difference between the calculations and the measurements after S.S.1. The measured high value of shape factor may be

due to flow near separation.

For both velocity profiles and integral parameters, an

appreciable difference between the present method and the

conventional one can be observed at S.S.1/2, where the

boundary layer thickness is not so small as compared with

the longitudinal curvature. It appears that the present method provides better prediction than the conventional

one.

Fig. 9 shows the pressure variation across the boundary layer at S.S.1 and 2. It is obvious that there exists appreci-able pressure variation which is against the assumption of the conventional boundary layer calculation.

The predicted pressure variation by means of the present

method which takes the longitudinal curvature effect into

account shows a tendency resembling the measured results.

3.3 Case of tanker model

The ship model is geometrically similar to the model

used in research program SR 159 of the Ship Research As-sociation of Japan. The body plan and principal particulars

are shown in Fig. 10. The model speed was 1.345 m/s, which corresponded to Reynolds number of 1.14 x 107.

The velocity measurements were made in full load

condi-tion at S.S.7, S.S.5 and S.S.2 as shown in Fig. 10. The

1.0 011 01 0.0.000 Measurement r1/,x le/ u, Calculation Present method Conventional method Cl) S.S.7 -U cocoa 0 50 (471m) i0 05 0I 000000 (2) S.S.7

me

0. 0 50 (run) 5 10

7

11,

12 iI)***

Fig. 9 Pressure variation in boundary layer

1,-8.000m 1../ B=6.000

B/d-2.760, Ca =0.802

Fig. 10 Body plan and measuring positions of tanker ship

model

calculated results are compared with the measured ones in Figs. 11 to 13.

Agreement between the calculated and the measured

velocity profiles of main flow is fairly well at S.S.7 and

S.S.5 except the bilge and the bottom at S.S.7. But agree-ment at the position near free surface and the bilge of S.S.2 is not so good. This may be due to the effect of free surface elevation and flow separation.

As for the cross flow, relatively large values were

meas-ured at the bottom and bilge part of S.S.2 and it can be

said

that the calculated values agree roughly with the

measured ones.

I. 0 0 0 0 o 0

510

(.mm)

0 Measurement

Calculation by the present method

05

OA{

0

Fig. 11 Comparison of calculated and measured velocity profiles (continued on p. 10)

WITB 133 July 1979 000 0 0000 00 A.! (ore) 50 4 7 V? 2 9 S.S.2 &SI

0.0.0

°

000 0 0

1, 0 Lf. o

"0°0 000

e 0 1[ 11° _0i (ID t_. 50 100 1 50 100 150 200 (mm) (mm) o lai c, 0

[

..0000 o o 0

0 00 °. 50 100 150 0 50 100 150 200 (mni) (rnrn) 0.1 01

0000

ob

0 , 0 000 0 ° °

00

o 0

0 00 "

°

i 50 100 150 0 50 100 150 200 (mml (rnm) 0 0 0.1 0' 0 0 _{0 0 0} 0 0 0 IL

MTB 133 July 1979 10 35 0! » I 5 S.S.5 UrF 0000 o ? 50 (mrn ) I. S.5.2 40 (mm) 0 100 0 0 0 0 yaf 1 1

I1

0 50 100 150 200 250 300 (mm) 0 0 o 0 o 0 0 0 o to. 0.5 05 ® S_S.5 0 oIU 0 u 0 0 -01 [ 0 . Trti 4111- ft 0 50 100 10 200 3 (rem) ioo 0 50 100 150 200 (mm) ® S.S.2 (00 1 0 05 S.S.5 S.S 2 . 0 - .-0.1

[

u 0

Fig. 12 Comparison of calculated and measured boundary layer parameters

0 50 100 150 200

3 (rem)

Fig. 11 Comparison of calculated and measured

velocity profiles 0 0 1 15 1.4 13 1.2 0 002 0.001 0 1.5[ 1.4 1 3 1.2 0.002 0.001 0 o Measurement present method conventional method Calculation by the Calculation by the SO 7

0H

o 2 8i, o ° 0 2 0, 0 0 0 90 -60 30 0(deg) 0 90

60

30

8)deg) 0 1.5 1.4 13 1.2 0004 0 003 0 002 0.001 S.S.2 28, 90 60 6 (deg) 30 0 0.1 Itr 0

00

10 tt/ U 05 S.55 100

010

0 o 50 (rem)

000

10 U 0 o o 0 10 @ S.S.2 05 90 0 0 0 mm -

oo

0 O

-o Measurement

Calculation by the present method

Fig. 13 Pressure variation in boundary layer

The predictions of the momentum thickness and shape

factor at S.S.7 and S.S.5 are fairly well but those at S.S.2

are not so good, as shown in Fig. 12.

In case of the cargo ship model, the predicted results

agree fairly well with the measured results before S.S.2 as

mentioned in the previous section. This indicates that it is more difficult to predict the boundary layer characteristics

of a full hull form than a fine one.

The difference between the two calculation methods is appreciable at S.S.2. The present method seems to give predictions somewhat improved in comparison with the conventional method, though the predicted values differ considerably from the measured results. The predicted pressure variations across the boundary layer agree well

with the measured ones as shown in Fig. 13.

4. Concluding remarks

Theoretical calculations of three-dimensional turbulent

boundary layer around ship models were done by means of

two integral methods. One is

the conventional method based on the assumption that the pressure is constant in

the boundary layer. The other, which was developed in the present study, is a kind of method of second order

approxi-mation which takes into account the pressure variation

across the boundary layer due to the longitudinal curvature effect.

The results of the calculation were compared with those

of the velocity measurements in the turbulent boundary

layer around ship models of different hull forms Wigley's form, a cargo ship and a tanker.

Agreement between the calculated and the measured

results was generally good in the forebody but discrepancy increased towards stern. Effect of pressure variation across

the boundary layer seemed to be appreciable in the

after-body and by taking this effect into account some

improve-ment was found for prediction of flow by calculation.

However, the discrepancy between the calculated and the measured results are still large near the stern where the

effect of separation is involved. To extend the applicability

of theoretical calculation, more radical approach is

con-sidered to be necessary such as investigation of the

struc-ture of viscous flow field at the stern and examination of

the basic assumption of the theoretical treatment.

U ; u, w, v ux, uy, u, MTB 133 July 1979 11 S 0.2 0

0

-0.2 -0.3 (-)So o°

-0 .-0 0 0 0 o 0.1 10%2 ° 0 o o 0 100 200 0 100 200 (mm) (mm) 0 o

0

°

o 0

o 0 0"...) 0 1 -02-

-

_{o 0} o o -0.3 2Co 0 100 200 (mm) (mm) Nomenclature B breadth of ship Cb block coefficient

Cp static pressure coefficient

d ; draft of ship

eN,

;

unit vectors in the streamline

coordi-nate system

e0

;

_{unit vectors in the (,}

_{0) coordinate}

system

e, eriy, enz ;

components of unit vector en

; = 61"10

H*

_;

_{= .51*)/0,i}

h1, h2, h3

;

metric coefficients of the streamline

coordinates

I,], k

unit vectors in the Cartesian

coordi-nate system

K,, K2; geodesic curvatures

K3, K4

;

normal curvatures PP ; length of ship

p ; static pressure

R ; radius of curvature of body surface velocity at

the outer edge of the

boundary layer free stream velocity

velocity components in the streamline coordinate system (X, 77)

velocity components on body surface

in an inviscid fluid in the Cartesian co-ordinate system

shear velocity Cartesian coordinates wall cross flow angle boundary layer thickness

(-0 u o

(1-0 )d.

c.) v

2*=JIJ

cic U

4

01

o U

u

e 12=f'; Tiv (1--)aT

UV

02 1=f

_{o U2}

US v-e 2 2 =

₀

U-streamline coordinates viscosity kinematic viscosity curvilinear coordinates density of fluid

components of shear stress in X and r? directions respectively - 100-flow, between, eg-,:

(1

-0.2 0:3 0,

-MTB 133 July 1979

Subscripts

w ; quantities evaluated at the body

sur-face

e ; quantities evaluated at the outer edge

of boundary layer

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