A streamline iteraction method for computing the three-dimensional turbulent flow around the stern and the wake of ship

CHINA SHIP SCIENTIFIC RESEARCH CENTER

Streamline Iterati

Method for Couting

The Three-Diniensional 'Iurbulènt PÏow Around

The Stern and in the Wake of Ship

Huang Shan

Thou Lii

May 1986

CSSRC Report

EngliSh version-86004

P. 0

.

BOX 116, WUXI,, JIANGSU

A STREAMLINE ITERATiON METHOD FOR COMPUTING

ThE THREE-DIMENSIONAL TURBULENT FLOW AROUND

THE STERN AND IN THE WAKE OF SHIP

(FIRST REPORT: WIGLEY SHIP MODEL)

S. HUANG and L-D. ZHOU

China Ship Scientific Research Center PO Box 116, WUXI, Jiangsu, China

ABSTRACT

A generalization has been made of the axisyinmetric streamline iteration method developed at China Ship Scientific Research Center several years ago. The pre-sent paper is concerned with the calculation of three-dimensional turbulent flow around ship, with emphasis to the stern region and the wake. In this method, all the Reynolds stress terms are incorporated into the basic equa-tions. Thus, the present method is more adaptive for calculations of the thick boundary layers in the stern region and the wake of ship.It is also shown that by using a coordinate transformation, the iterative matching

be-tween viscous flow and potential flow can be avoided.

Numerical calculation is carried out for a Wigley model to test the numerical solution procedure. The results of ca2culticn show reasonable agreements With

the measured ones. Further calculations will be made for ship ofan arbitrary form.

NOMENCLATURE

A Cros s are a of s t re amt ube

a Constant in coordinate transformation

Cl,C2,CD Constants of K- mode1' turbulence

C, Pressure coefficient

C,C

C1-.eracteristic lengths

K Turbulent kinetic energy

L,B,D Principal particulars for ship model

Mixing length

P Static pressure

P Total pressure

P, Free-stream presaure

Q Mass rate of flow

c1 ,q2 ,q3 Nonorthogonal curvilinear coordinates

Transformetl rnorthogonal curvilinear coordinates

Re Reynolds nimber

Cylindriai polar coordinates Distance ft the wall

S Distance eadi3g edge

u1,u2,u3 Velocity copØelts in CartesLan coordinates

u Velocity eross the boundary layer

z-V _{Absolute value of velocity vector}

V,V6,V

Velocity components in cylindrical polar- coòrdinate

V Free-stream velocity

Cartesian coordinates

.

-Coordinates of-parameter plane BOundary layer thicknesses

y Angle made by

èridonal srelrie wth-Z-aixs

Dissipation rate of. turbülent-energy Coordinates of

transformation

eff'»l?t ,u Coefficients í visáosity

-auj u Reynolds stresses

Density of fluid

eff.k,aéff. Constants of Xmodel. of turbulence

1..- Stress tensor in Cartesian coordinates

Trr,T.e,Trz,etc. Shear stresses in cylindrical polar o6rdinates

A _{gene-ralflowvariable.}

Coefficients in isoparamatric. transföi'mation equations Angle macle by-meridional plane with spacial streamline

s

- Reia,-atior factor

TNTROUCTIOÑ

In recent years, considerable achievements h-ave been.made -in the field of calòulatiori o _{the thick boundary layer over the ship-stern- and in the wake,} which -now becomes the essential for ship désign and reduction of vibration

ex-citation and noise cThe tc Dropéller-hull interaction. Since the SSPA-IrrC Wòrkshnp on ship Eound&ry Layers held at the Swedish- M-a'itime, Research Center

SSPA in June l-98Ornyr.umerical-methods have been developed, ranging from the first c'der boundary layer- theory to fully elliptic solution.' The nüweric-al experiments indicate that nüweric-almost nüweric-all presented-method- gi1e good results on -the most part of ship hull. It is also revealed, however, that the first _order

boundary layer theOry does not produce -a precise picture. for the flow over-ship-stern, since- in thát region the rapidly thickening boundary layer _causes the assumptions embodied in the theory to -breakdöwn. -

-Among various - approaches for thick bouñdary layer, the most prevailing method is the so-càlled partially-parabolic flow solution- procedure pursued by Spald--ing and Mur'aoka et al. see -Peference _{[lj-- (i). Except tite coordinate system,} all of these reporté are essentially the same as such bäsed on the K- mode! of turbulence and the partially-parabolic equations Obtained by neclecting some terms in -the f1ly elliptic Reynolds equations. -Besides the basic -assumptions adopted in partially-parao1ic solution procedure arid the

cooi-dinate system -whicÌ may iñfluence the accuracy of caiculations, another weakness is the- negligence, of viscou-invicid interaction which _{may also}

causes inaccüracy in solution. - -

--Recently, a very promising development of the p-ati-allyparaboiic- _{-solution has} been made, by (Then nd Pat-el

f83

_{üth numerically-genéràted, body-fitted}

-coordinates and..the finite-analytic technique.

Other paper of interest is the

one reported by Larson

and Johansson [ 9

J ,

namely the streajiiline curvature

rrethod, which is. similar to the present paper

in some respects.

At China Ship Scientific Research Center, a dife'ent approach, the

stream-lineiteration method, has been developed by

one.of thé present authors,.ee

Zhou (io)

The excellent calculation results of this method for the turbu'ent

flow over the tail and in the

wake of an axisyminetriC body encouraged us to

mäke further effOrts for

generalizing -it to the calculation of turbulent f lo

rounc1 three-dimensional surface

ship, which gives rise to the present paper.

The important features: of present work are summrized here. First, no

assump-tioris are made' for solvinp the Ryno1ds

equation, i.e. the fully elliptic

equations are employed. They are solved exactly in a three-dimensional

non-orthogonal curvillnéar coordinate.sySteln

with the well-known K-.c model of

turbulence. We are convinced that this

nièthod is more adaptive for the calcu]

ation of thick boundary layer in ship stern

region since it contains all

higher order .effects. Secondly, a coordinate transformation is used

totrans--form the flo

field which extends to infirdty in both' longitudinal arid radia].

directions into a fihité regién. It is not necessary to

perform the

viscous-inviscid matching since the boundary layer flow

and the flow outside the

boundary layer are détermined by an

uniform equation system. The boundary

conditions for the transformed solutiçn domain can. be given readily and

accurately. Finally,.a1.lthe basic equations are

changed into theconvenient

fôrms for streamline iteration calculations.

However, tie location of

mean-flow streanlires is unknown beforehand and must be

determined by an iterative

scheme..

As it will be seeñ later., this extension work

is not as simple as it looks

like. Apart form the basic equations which are thore

complicate .in

three-dimensional prcblein; there are sevetal difficulties to be:

overcoined, .fr

example, the assurplicn of iritial mean-flow streamlines

and the determination

of the locations of streamlIneS in the. ite'ation.procedure.

As the emphas

is placed on the solving of the latter, the calculation

is applied to a body

of comparative

simple geometrical form, namely.the Wigley4oubie model. The

éompáriso1S of numerical results with experimental data are presented

in this

paper., and agreements between them are

satisfactory. It has been decided to

make further efforts for the application of this method to ship of arb trary

form.

. . .

2. OUT.1.INE OF THIS !OETHOD

2.1.. Governing Equations

For the t1ree-dimensionál steady. and iÍLcomressible flow, the governing,

equations for time-averaged velocity components

and sta-tic pressure in

Cartesian coordinates are given in tensor notation as follows

Continuity équation

. . - . . ,. -.

-3--- (.)G (i1,2,3) (1) x. i

Momentum ecuationS

T (Puu.) _- +

(t)

(i,jl,2,3)

(2)

For the turbulent flow, the stress tensor in equation (2) is given by

T...(

+

Under the concept of effective viscosity, the tensor is defined by

i ( )

eff 3x. 3x.

- J i

where

eff' so called effective turbulent viscosity, is given by (see Harlow and Nakayama 11] , Launder arid Spalding [12

J

eff1 + 'ti

+ C0PK/E

The governing equations for K and c are described as follows:

3 3 e'ff 3K

-

(puK)--0effk

) + GE

--.1 (puc)--- (

af

.!_ _{+ C1GE} - C2p .-where GE is given by GE

₄

+ .j. )

pi

ifl

equations (5), (6) arid (7), there are five empirical constants C1,

'2'

Cçff.}

and 0effc The values of those constants are given in 1.

Table 1

Empirical constants for K-c model

C1 C2 _CD

0effk

0effc

1.'4 1.92 0.09 1 1.23

Equations (1) and (2) with (4) through (8) are closure for six unknown variables u. ,u2,u3,P,K and c. Obviously, it is very difficult to solve this nonlinear equation system directly. With the difficulties in mind, the numeri-cal method should be carefully chosen.

2.2. Coordinate System and Corresponding Ecuations.

In the present paper, a cylindrical polar coordinate system (r,e,Z) fixed to ship is used, where Z-axis is positive towards the stern, as shown j. Figure 1. The water plane is defined by Oe while the vertical plane is defined by

¡t the sane time, a nonorthogonal curvilinear coordinate system (q1,q2, q3) is emp'oyed, where q3 lines coincide with the mean-flow streamlines and q13q2 linesforirt a two dimensional nonorthogonal curvilinear system in every

section, as shown in Figure 2. The first line q1constant coincides with

the

or c'rtour

uf

ship in each section. P] the flow variables are

-tcried át fr,ite difference grid nodes formed by the lines q1,q2

arid q3.

However, the locations cf these grid nodes are determined in the cylindrical

polar coordinate system. Ps the mean-flow streamlines, namely the q3

lines,

are not known beforehand, the determination of the coordinate system

will be

a part

of

solution. Art iterative scheme is used which can revise the

initial

assumptions made to the mean-flow streamlines step by step until the f low

field has converged within a prescribed tolerance. Owing to the feature of

streamline iteration method that the grid nodes are located on the

mean-flow streamlines, the above basic equations are transformed into forms

which

can be conveniently solved in present coordinate system (for more detail

informa-tior., see Zhou E

10] )

Continuity equation

Çjeti.dA Q

constant

(9)

where A is the cross area of streamtube. iiergy equation dp01.

jjifB(rrrr)

I13 dP

¿pI_

if(rt4

kT

1 er

err,

eOEcre)i BO +

ez

J

r

+

i!!4

e(rre1)1+Z,,t

e? J.y)

er,? a(rl)

e?

r,,

e J r

e[vv+

YL.-Y-T-+Vì-YL--1V.2]+

rae

r

f

r e(rt)

et,

a(yCr,) tè6 lI

iL er

ai

+ a

j Y

ar4YW

-.5-(10)

(12)

41

(rt,.,)

eti

e(r)

+ tri

r

[y

e V. a V.

ç +

.íJ

}

(

er

5W-4. a

-ar..

(rte?)

er

ai

+

er[v"

V.Wo

VeVrJ+

r

03)

a

r

_{JJ dGI%}

where

the total pressure F

is defined as follows:

P0 = P e

te =

i

( 14H

E \

+ '

ìi.4L. a

-

(-4ft

ar)

WaOI 0 J

L1('L\J

+

E-?ae61J.Koe J

Wt«ÇICfl

a._i

d+

ALi d9

-4

I d y

a+....

ij.L

d+

dr

W

, dG (f

dr

T

¼ dr (16.). (17)

=4

_f

,'.4..

_ni

(i

ve +

u? 'r

7

In t-ìis paper., the symbols

.

and denote the di'ectional derivative operatorswhich wean the derivätives with respect toZ, rand ê along q1 line, resr.ectively.

Equations (9) through (16) ar bic equations- of presnt numerical method. they are ccsure for solving the three-dimensional turbulent f Ïow. As men-tioned äbove, thd flow variables ar given at the grid nodes formed by lines q1,q and q3 Therefore, ti-e partial derivatives of flow variables with respect tt i,6 and Z appeared in the right hand sides of equations (10) through (16) can not be evaluated easily with numerical differential method !athematïcal technique should be applied.

Let denote a certain flow variablè. Then,acccrdjng to the role cf direction-: al derivative, the follôwing relations may bè easily obtained:

1(.cnuation

dW_t

ar'

j4

ar

e equation

-where t-n.=

r

_"ru

j1(+L-

_'ar

_rae

r=-"li (1L

at

where, is given by .

-41-

(20)

Thus, with equations (17,) through. (2.0), all the partial derivatives can be replaced by directional derivatives al6ng q1,q2 and q3 whiàhay be computed by numerical defferenti a1 method.

So far, ail the necessary equations havè bç.n obtained. With sore boundary conditions, the numerical evalution can be performed iii principle. To overcome the diffiòulties of determining the outer and exit boundary locations and the conditions on them, a coordinate transformation technique is employed.

.2.3 Coordïnate Transformation

The determination of the boundary conditions, epecially the outer and exit boundary conditions, is a tricky job. The common practice is to cut off the infinite region with finite boundaries and deterine the conditions on them -by the calculation of potential flow or by ari iterative viscous-inviscid

matching procedure. In order to overcome the weaknesses aroused froir this practice, a coordinate tranformation was developed in our previous paper

CJ , can transfo*w the infinite flow region intoa finite regin. As a result, the free-stream. conditiohs and the parabolic flow conditiis can be employed at the outer and exit boundaries, respectively. On otherhand Chen

and Patel C 83 chose a large solution doman which occupies the entire zone öf viscous-inviscid interaction and the free-stream conditions are directly used on the outer boundary. this technique is very prcticai. But it breako the perfection of theory.

In this paper,.the coordinate transforration ex'tended from our previous work

is expressed as follows: .

-nC.

ìì=l-a

r

e

arctg

where, C and Cr are lonitùdinal and radial characteristic lengths.,

respec-tively. The constant ais gréater then 1.

-All-equations of (10) through (16) are solved in the transformed domain with grid nodes formed by the tranformed curvilinear lines -qjq ándq. Before solving the equations, some relations between the denivatives in physical spaeand in transformed space must be obtained.

uieth, the f

derivatives in the transformed space such as

1q

_'

1qj ''q;

'

Iq

'

Iq

_'

and

cari be evaluated.

Then, under tJ

rcle of derivation for compound function, the following

ieticns c

easily obtained.

¡1

f

(22)

q3 q3

-

4:Ji

_ma

₍₂₃₎

dr

_q1

-

_r

_q1 I 1 2 ii

ç

I q3

dei

_cos2

lT dÇl

al

_q3

--

CZ

_tlt

_q3

dr!

2 C 1

_dnl

de1 q2 q2

_iil-ii

d

d'

-

-- ma

-

r

q1

As the coordinates ri,

and

of the grid nodes arid the values of flow variable

4)

on these points ere given, its directional derivatives, and subseouently

the partial derivatives in physical space can be evaluated with equations (17)

through (20) and (22) through (28). 'ith the free-stream conditions on the

outer boundary and other boundary conditions, the calculation can be started.

2.L. Boundazy Conditions

The boundary conditions at each boundary of the solution region are defined

as follows:

Hull surface

Vz

Vr = V8 = O

and K==O

Outer boundary (infinity in radial direction)

y

= y

- U

z 3!<

3r

3r

arid

(2'4)

del

lT q2 q2

dri

2

Cr

1 Iq3

-

i

Water and center vertical plane V

e 'ae 3e

and

3K_3E:

-. =

(U) Exit plane (infinity in longitudinal direction)

=

I=

O

and 3K

- -.2-

-(5) Inlet plane

The inlet plane is located at the middle of ship model. The boundary thickness in this plane is calculated by Schlichting's formula for flat plate (see Schlichting 1 13) )

o = O.37s u

-1/5

(29)

where s is the distance from leading edge. For the longitudinal, velocity V, the power law is used while thé velocities Vr arid Ve at this plane are forced to be zero. In the case of pressure P, it is provided from the calculation of potential flow with tne method of Hess and Smith tlUJ

Input data for turbulent energy K are suirired up from the experimental data while Bradshaw's empirical profile is used for the input ofmixing length l (15] . The value of is determined from the 1< and the l as follows:

C

CK/

(30)

2.5. Details of Numerical Process

As mentioned above, the present numerical

method

must adopt an iterative scheme. Some steps follow simple 2-D geometry case which have been described in details. in cur previous paper [10 ₃ . Theréfore, it is appropriate to state them briefly here. The solution algorithm follows the next steps.

Assume the initial locations of mean-flow streamlines in the flow field. A good distribution of initial streamlines, which is dependent ori the

physic-al concept of flow situation, not only reduces much computation time, but also guarantees the convergence of nuinrical procedure. A general procedure for this step will be accomplished in near future.

For V, K and E, the initial distributions ori each station are assiimed.tc be the saine as those at the inlet plane station.

Calculate dr

d1

and drI t from the coordinates of grid

nodes by nun;erical differeritia] method. Then, the distributions of V,, and

can be Obtained as follows:

dr

2 1

2( n

Cr

1 dr - ( - V

-

ç

-

.I T z

t

i

31)

q3 q3 de

Cr 1(1fl)

.2 W d

- Vr

- -

COS ( ) V

(32)

q3 q3

(3) Compute the first partial derivatives of Vr

V8 andV with respect to Z,

r and 8.

('4) Integrate the K and e equàtions (14) and (15) to obtain new distributions

of K, e arid Ueff!

Use the new distribution of

_4ef

to obtain the stress tensor and

com-pute, the ñecessary first partial derivatives of stresses.

Integrate t:epressue

radieñt equations (12) and (13) tc obtain a new

static pressure

at a]1 grid nodes ir. the entiré flow field. The integral

constant for the equation (12) is determined by the bounda, condition P:P

at n. while for the equation (13) it is déermined. by the solution of

equa-tion (12).

Iitegrate the enerFy equation (lo) to get a new distribution of total

pressure P

Compute the new distribution of V

fron' folläwing formulae:

v

j2(Po.-P)

v/Á+te

A:+tg2

dr

f2

1 2 7T

Cr-

1. d

where y: arctg

_):arçtg1

cos

i. z q3

arctg (r

(

/iit,2y

) q3

Cr lri(l-n)

Cz Tha

= arcte

- lo -.

)

q;///1+tg2Y]

]

(36

(9) Determine new distribution of the mean-fÏow streàmuines, in such wáy that

the contiiìuity is maintainEd within each streaistubé. A difficulty

encountered

here is similar to crie re.perted by Larsson and Johansson [91

, namely the

Lateral dis1acement of the streamlines. In two dimensional problem

as our

previous paper, there will be no prdb1en' about it since the

ma

rate of flow

at each stätion is a nionotonic function of a single variable. Py inverse

interpolation of moss rate of flow, it is easily to determine the locations

of

str'eaniiñes. In three dimensional problem, hùwever, the

mass rate of fld

at

-every station is o functionof two variables, fcr éxarì1e, the r and P incur

with continuity equation is to

Solve an indeterminate equat-ion..

_Severml

approaches were developed for this pràb.lem,

_{see Xin and}

_Jiang

_[16

1

Ison

and

-Johansson

[j

In this paper,

_añother

_{approach of'two}

steps is,.

develop-ed which embodies some ideäls of finite

_{elemrit méthod.}

(a) Isopa±'ane±rjc transformation

Assume ABCD is acurvilinear

_{quadrilateralin (r,8)}

_plane,

_{as shown in Figure}

3

(a). hé coordinates for the four

apexes are given as fàll:

A(rkj

_0k,j

_Bk,j+l

_0k,j+])

C(rk+lJ+l ek+lJ+l)5 D(rk+,J _8k+l,j

The equations for the four ié _{lines are supposed to.b}

as follows:

AB:

_Oq(r)

_CD:

AD:

r:q(e)

_BC:

rq1(.e).

Ç3 7)

A

transforiation.

_{is sought to map the}

curvilineär quadrilateral ABCD into * * **

a rectangleA B C D _{in a paramcter plañe (a,ß),}

as shown in

_{Figure 3(b), and}

satisfy fcJc'wïng conditions:

A B : ß:B.

;.

C*D:

AD :

_BC;

(38)

In the

_{eaSe. of}

four

_{side lines of ABCD are straight segments, a fourpoints'}

.soparametrjctransfoatjon is enough for _{the accurate}

_{mappIng. Obviotly,}

in general case the four side lines are curvilinear _{Thus, the error due to} the straight segment

approximation is considerable esPecially when the curvi-linear quádriateral is large. To givé a more accurate _{transformatjcn, the} grid nodes in Çr,e _{plane are doubled by quadratic}

interpolation andthe

points

E (rk+1J

_{8k+-,j' F (rk]3, 6k,j+'}

(rk++l, Rk+,)+l)

and

'k+l +'

_'J

₂

_8k+l

_'J

are obtained. Ther., thé mapping is _{perfòrmedwjth à} . .

eight-points isoparametric_{trnsforniation described as follows:}

(see

Pig.4).

(

r

cm 6

Lj

;,0 SinO

r

_{[ (}

cs 8 _+P;j .

1.j+ CS

+

r:.

c-os +1fr4 Lj4

₊

_;«;ji1 Cø e14 _+K4,f11 4,jtk +

cs

+

j4 +X,jf

+

11

-'*

'g 4'iIz.j49+

ßtrt j41 %C4.sjkl

+

I' 6KIt4

c+o.j + cm + tn

*

)Z

0=

orctj[(

rK + Çin

0.pi4's.j+I

r44

+E41,j++ * $lflO44 IC41 +rE+,

"8K*j

+%ij )/

( ç,, c.s 9 + rK14. GOS

rK,,I

9 +

CaS

_{i+,j+I +IC+l.I +}

r,1d CaS OY1(j44 +K41,jtl

l

.

c'c9'.

_*

j J where 1'

=('°") (- ?')

I)

=

(i4') (:-p)

(-.'.-i)

4K44,j4

_(i_p(i+ø')

=

-(''.)

(j1g*)

(j)

=

-(i°'')

(i+9i)

=

j(jØf&) (,+p*)( pci)

=

£(_p13)(i

I

₎

-where

Gc.*= (2oC,.,)/(«K.c'k)

p(2Pfl,&H)/(flJwii)

By these equations, all the meshes in (r,e)

plane are

transformed into the parameter plane. Then the inverse interpolation of mass rate of flow is

ter

foreed in the parameter plane to find new locations of streamlines.

(b) inverse interpolation of mass rate of flow.

This procedure may be described with reference to Fipure , whici' exriain hc

12

the new location for:an arbitrary grid node A* on cer.tain station rilane is determined. First, it is supposed that no mass räte of flow passes through the lines S, S and S since they are considered as the intersections of stream-surfaces with the station plane, as shown in Figure .5(a). Secondly, with the distribution of velocity at this station plane calculated fron' step (8), the relation between coordinate and mass rate of flow across the strIp zone bounded, by the lines S and S can be obtained, as shown by the curve in Figuré 6(b). Then, accordinE tò th.e prescribed distribttion of mass rate of flow across the corresponding strip zone at the inlet station plane, ar

in-verse interpolation of the mass rate of flow can be performed and the ne coordinate. aGl for A* is obtained, as shown in Figure 5(b). However, as a-pears from the Figure 6(b) and Cc), there is a deviation-between and

G2

which is. obtained in the same way as except that the strip zone is

bound-2- 3

ed by the lines and The mean value of aGl and is adopted for the a coordinate' of A*

+ cZG

By repeating the sanie. procedure 'at the ß direction, the new B coprdinate of

A*' can be òbtained- .

G1 +

By substituding cx, arid into the right hand sidesof equations (39) and (4C)1 we can get a set of coordinates (r1f81f). Finally, the new coordinates, for are obtained as follows:

"Anew rAOld + w(r.f - "Aold

(L5)

Anew e?Old + W(61f - eAOld) whére

Acid and 6Aold are coordnat'es of A at the pre'rious .teràtion.

(10) Repeat step (2) through (9) until tie ntaxintuin 'deviation between' the loca-tions of the streamlines in two successive iteration is within prescribed accuracy. Then, these distributions of VrVOVZP K and are the final xesults of the problem..

3. NUMERICAL EXAMPLE

The: validity, of the present solution procedure is testec 'by a nuirerical xample. The calculation was carried out for the..Wig'ley model measured 'by Hatoand Rotta f17) . The fárm of this model, as shown in Figure 6, can' be analyticaLly

represented by ' .

B

{.

2J[

( )2]

' (175

The principal particulars of it are given in Table 2.

-Table 2

Princpa1 particulars of the Wigley ship model

The calculation is started from the middle of ship. The mesh points in _the

(n , _{, E) directions are (27,12,22). A 27x24x22 grid nodes distribution was}

also tested, and no significant improvements were made.

The measurements are carried out along the normals which intersect with _four streamlines cf potential flow on the surface. The four streeîrlines _{are named} as streamline no.1, ... , streamline no.4 according to the order from the keel line to the.waterline, as shown in Figure 7. For the streamline _{no.1 two kinds} of normals are adopted. One of them is a normal taken at the, surface _nearest to the cusp and another is one stood throuph the cusp in the dirction _{of X3} axis. When the morirai in X _{direction intersects the streamline no.1, this}

D

line is named particularly as streamline no.0. We shall present the calculat-ed results along a1 _{these normals except those along streamlines 1,2 and 3} at. Z/L:O.9,O.9E,O.975 and 1.0 since a considerable amount of interpolation is required fcr the comparisons betweén calculations and _{measurements.}

Ir. the fpures 8-12, calculated mean velocity profiles across the boundarr layer a in the wake _{re compared with measurements. The abscissa rD/s is the} fraction of the distance from the wall _{or wake centerplane to 'the measured} boundary layer thickness. _{iri general, there is good correspondence between} the calculations and measurements.

In figure 13, the pressure distributicns _{on the body surface in the two} symmetry plames are presented. No experimental data of these distributions have been published.

In figure 14-17, the calculated and the measured results of turbulent pro-perties along two symmetry planes, i.e. turbulent kinetic energy K and mixing length are given. In some figures, only the calculated results _are tresent-ed, since no experimental data have been published there.

In our numerical experiments, the discrepancy between the fully-elliptic and the partially-parabolic solutions have been studied numerically to show the limitation of partially-parabolic approximation. It is revealed that the dif-ferences between them are very small irs. this _{case, which perhaps is due to the} slender hull form.cf the model and the comparative small Reynolds _number. Further studies are required in this aspect.

4. CONCLUSIONS

(1) By the development of our previous work, _{a numerical solution method for}

-14-L B D R

e

prediction of turbulent flow around

_{the stern of ship and in the iake is}

presented here, effectiveness of which is.

_{domenstrated by numerical exarrple.}

In general,

_{he agreements between the calculated aiid measured results. are}

.satis factory.

The coordinate transfor'matjon

_{permits us to take the free-strean}

_conditions

as the outer boundary conditions

_{In this way, the accuracy of}

_{calculation is}

improved ánd the cómputation c°st is

_dro,ped.

In the symmetry plane where the ship nodel has a sharp edge.,

_{the well-known}

symmetry condition should be

_{riven special consideration}

_{Also, experimental}

information. on such piane needs

_{to be examined from various perspectives.}

(Li)

_{Considerable more work is required}

for the ful], generalization of this

numerica]. procedure. Work will be

_{continued to extend the present method for}

the calculations of flow around ship

_{with an arbitrary frm and}

_interaction

of. a hull and a propeller in

_operation.

5. ACKNOFYLFCEMENT

The authòrs are indebted to Gu Cheng-hor1g

an

_{Wang Ling-di foi' their.}

tyewriting and plotting.

REFERENCES

[li Patankar, S.V. and Spalding,

_{D..B. :"A Calculation Procedure for Heat, Mass}

and Momentum Transfer in

_{Three-Dimensina]. Flow". mt. J. Heat Mass}

Transfer,

Vol.. 15, 1972

[.23

Sp'àlding, D.B. :

_{Calculation Procedure of 3-D Parabolic.}

_{and Partially}

Parabolic Fl-ow". Imperica]. Colledge

_{Report HTS/75Ï5, 1975}

[33

_{Muraoka, X. and Shirose, Y.:"Calcu].ation}

of.SSPA 720 and HSVA Tanker".

mt. Symp. on ship ViscOus Resistance,

_{SSPA, G6teborg, 1980}

[4)

_{Abdimeguid, A.M., Marlcatos, Ñ.C., Muraoka, K and Spalding, !D.B.:"}

A

Comparis

_{Between the Parabolic and}

_{Partially-Parabolic Solution Procedure}

fo±Three-Djmsjonai TurbulentFlows

around Ship's Eul]s

.

Appi. Math.

Modelling, VOl.3, 1979

15)

Muraoka, K:

_{Calcu1atjon of Thick Boundary Layer and Wake of ship by}

a

P.artially Parabolic Method

.

Proc. of l3th.Syip. on Naval

Hydrodynamics,

Tyo, 1980

. V

t

6]

Muraoka, K

_{Calculat3on of Viscous Flow}

_{around ships With Parabolic}

and

Partially Parabolic Flow

Solution Procedure

.

Trabs. of West Japan Soc.

_Nav

Arch., Vol. 63, 1982

15

[71 Tzabiras, G.D. and Loukakis, T.A., A Method for Prediction the Flow

around the Stern of Double Ship Hulls . mt. Shipbuilding Progress, Vol.30, 1980

(8] Chen, H.C.

and

Patel, V.C.: Calculation cf stern Flows by a Time-Marching Solution of the Partially-Parabolic Equations . 15th Symp. on Naval

Hydrody-namics, Hamburg, 198L

[9) Larsson, L. and Johanssor1, L.E.: A Streamline Curvature Method for Com-puting the Flow Near Bhip Sterns . Proc. of 1Lth Syirp. on Naval Hydrodynamics, Ann Arbor, 1982

C].o1 Zhou, L.I.: 7A Strearn1ire Iteration Method for Calculating Turbulent Flow

around

the Stern of a.Body cf Revolution

and

Its Wake . Proc. cf 1Lth Symp. ori Naval

Hydrodynamics,

Prn Arbor, 1982

[ll],,Harlow, I.E. an abayarna, P.T. : 'Transport of Turbulence Enervy Decay Rate . Los Alanos Sci, aD. , Univ. California Feport LA-38514, 1968

T

t12] Launder, B.E. and palding, D.B.: Mathematical Models of Turbulence Academic Press, 1972

(13] Schlichting, Boundary Layer Theory . Mcraw-Hil1 Book Company, 1979

[114] Hess, J.L. and

STrtti,

P.M.fl. Calculation of Non-Lifting Potentia Flow about Arbitrary Three Cimen:on Bodies . Douglas, A.C. , 1962

[15) Bradshaw, P., ferrs, ..U. anr ttwell, N.B.: Calculation of Boundary Layer Development Using the Turbulent Fquation . J. Fluid Mech., Vol. 28, 1967

(163 Xin, X.

K. and

Jiang, J.L. : The Arbitrary Qusi-Orthogonal Surface TMethod for Computing Three-Dirrensicna Flow in Turbine Machinery . J. of Mech., Vol.?,

1977 C

in Chinese)

(17] Hata.no, F. and Motta, T.: Turbulence Measurements in the Flow arounc a

Body ir a Circulating Water Channel . Trans. West Japan Soc. Nay.

S'4, 1982

-Fig.

I

The (r,ø,Z) coordinate

_system

Fig.

2 The

13q2,q3) coordinate

_yStew

17

ß.j

B

Fig. 3

FOUD_poifltSiSCPramet

transfcrmatìO1

I

-Fig.

'e

Light-pO'itø

opaiantriC tÑnsfqx'inatì-0fl

18

-*

C

B*

C

a'-'

(o

Q'

(C)

Fig..

irvere interpolation of mass rate of flo.i

°G2

19

-.0

xl

FIg .6. wgy ShiP

Mod

-o

L X? .1 ,2 3 2 2 2

C.-

s-A Zu..

0.9

0.95

0.975

rD

---

--f

r

r I I i i I I I _i I I I r D 1Streamline no.0 Streamline nc.1 20 -Streamline no.2 Streamline no.3

--St=in flQJL

1.0 1.05 1.]. 1% Stream1ine S I ¿rio.0

_,

O

t

flg.7 The nora1s and four s'treamiines(scale:1/2).

no.1

00

0009

0 0

0Q

000

o c

-

Z/L=o.9

o

o-

O

/L:o.9S

/L:o.975

z / L 1.0

o

Streâmiifle no.0

Q ExpeDieflt

CalÇulàOfl

1. oS

/L:i.1

Fig.8

Distributions-. òf velocity across thé boundary 1aers

-.21-.

-

I-1.5

u

V

.1. o

1.0

Lb' - - 1.5

r/

Fig.9 Distributions of vélòcity across. the bondar layer

1.0 0.5 - 1.o.

StreawHne nc.l

O Experiment

-

Calculation 1-ç

Z/L:l'..05

ZIL-rD

.ig.1O DiStributionS of velocity àcrñrthe boundary -layers

22

-Streath1ire no.2

O Experiment

- Calculation

s

&

0. 5

ö000

Q

. 23

-1.5

Z/L=c.25

Z/Lo.975

Streamline no.T4

O

Experi?flt

- Calcul&tiOfl

Fig. 12 Distributions of velocity across the boundary layers

-0.O'e.

0.5 1.0

Z, L

Lo

Fig. 13 Pressure distributions along 0:0 and 8:ir/2

Cp P/pV

24 -0.0L Z/L.:l.05 Streamline no.3 O Experiment - Calculation Z/L LO

0.0e e=o 0.OL4

8=ir/2 0.0

00

cP

XI

VX

1O

D"6

Fig.l'e

Turbulent

kinetic energy distributions along 8;0 o Experiment - Calculation

25

3

K/VX103

3

Z/L1.0

Fig.I5

Tux'bul.ent kinetic energy distributions along e/2

o fxperiment

- Cajculotion

-26-rD/

0

0.2

r/6

2_

i

o o o o

_{0 0000}

r

0.2

Z/L= 1. 05 o o o e o

000

000

I I I

0.2

0.06

0.

QL

0.02

1 /6

m

o-Z/L=1.].

0.06

0.0L

-0.02

6

0.2

o

0.2

0 0.2

r../6

0

0.2

6

F1.i6

rnixing 1enght distribtticns

1orig &0

Experimert

- Calulation

27-i 16

_in

ZIL0.97S

0

0.2

tg.17

?.ixin' 1r.ght distributions along e:,t/2

o Eperiment

- Caicuiatior'

28

-DItS

0.06 -

0.06

o.01

-

0. 0e

0.02

0.02

-o J. ¿ o

0.2

0.06

0. 0L

0.02

ZIL:l.l

C.06

-o. o

0.02

r

0.2