A streamline-interaction method for calculating turbulent flow around the stern of a body of revolution and its wake

CHINA SHIP SCIENTIFIC RESEARCH

CENTER

A Streamline-Iteration

Method for Calculating

Turbulent Flow Around

The Stern of A Body

of Revolution and

Its Wake

Zhou Liandi

December 1986

CSSRC Report

English version-86011

(Presented at the Fourteenth

Symposium on Naval

Hydrodynamics.

August 23-27, 1982)

P. 0 . BOX 116, WUXI, JIANGSU

CHINA

A STREA'flINE-ITERATION

METHOD FOR CALCULATING TURBULENT FLOW AROUND THE STERN OF A BODY OF REVOLUTION AND ITS WAKE

Zhou Lian-di

China Ship Scientific Research Centre Wuxi, Jiangsu, China

Ab str a c t

This paper presents a new numerical method for calculating viscous flow around the stern of a body of revolution and its wake using a two-equation (K-E) model developed by Harlow and Nakayama (1968) and mod-elled by Launder and Spaiding (1972). The features of this method are: 1) The grid points calculated are taken on mean-flow streamlines and on radial straight lines, thus the convection terms of the governing equa-tions for total pressure, turbulent kinetic energy and its dissipation can be written in the form of their variations along streamlines. For

static pressure the radial pressure gradient equation is used. _These equations arc convenient for numerical calculations. The mean-flow streamlines, however, are not known beforehand and must be determined by an iterative scheme. 2) By means of a system of coordinate trans-formations, the calculating region is extended to infinity in both

ra-dial and axial direction. By doing so, the free-stream condition and the parabolic flow condition may be used at the outer and the down-stream boundary respectively. The flow in the boundary layer and the potential flow outside the boundary layer can be determined by an uni-form equation system. 3) Assumptions for a thin boundary layer and partially parabolic flow, etc., are exempted.

The velocity profiles, the variations of static pressure and tur-bulent properties calculated theoretically at some axial stations are compared with the experiments by Huang et al. (1978), the agreements being satisfactory. For the wake, the accuracy of the present method

NL1ENCLATURE

Alphabet Symbols

C1, C2, CD ; constants of the (K-c) turbulence model

C _{pressure coefficient (p-p)/ pV2}

f

; distribution profile of K

g ; distribution profile of Q

K _{turbulent kinetic energy}

2. _{distance along spatial streamline}

unit vector in spatial streamline direction

2. mixing length

L _{length of a body of revolution}

in

; meridional streamline, distance along meridional streamline

p ; static pressure (time-averaged)

Po ; total pressure

p+

v2

p ; free-stream static pressure

q ; arbitrary spatial curve, distance along an arbitrary spatial curve

Q ; mass rate of flow

r, 8, z ; cylindrical polar coordinates

r ; local radius of a body of revolution

r _{; maximum radius of a body of revolution}

max

R _{Reynolds number}

e

-S

; distance from the leading edge

u,v,w , axial, radial and circumferential velocity

com-ponents in a cylindrical polar coordinate _system (time-averaged)

velocity

meridional velôcit.y

/u2+v2

free-stream velocity

axial coordinate

Greek Synbols

ci ; angle made by meridional streamlines with the i-axis

boundary layer thickness

c ; dissipation rate of turbulence energy

coordinate transformations of cy coefficients of viscosity p ; density

ef

; Prandtl/Schmidt number , erf,c stress

a general fluid variable

relaxation factor used to modify the location f streamlines

y

;

V ;

V

;

A STREì'nJ::E-ITEF;TION iETHOD FOR CALCULATING TURBULENT FLOW AROUND THE STERN OF A BODY OF REVOLUTION AND ITS WAKE

Zhou Lían-di

China Ship Scientific Research Centre uxi, Jiangsu, China

I. INTRODUCTION

Theoretical predictions of flow field around ship stern and its wake are among the most inportant problems in naval hydrodynamics. Tn

order to reduce cavitation erosion, vibration excitation and noise due to propeller-hull interaction and to prepare guidelines for aft-end design of ships, it is considered necessary to predict the flow around ship stern and its wake. :iany research workers are engaged in this work. Among the methods for calculating turbulent flow around ship stern, the numerical method adopting two-equation turbulence model (K-c) and partially parabolic flow assumption is the most effective and popular one at present. As mentioned by arkatos and Wills (1980), however, there is a serious limitation in the above method, namely, the necessity of prescribing the locations of the outer and the downstream boundary and of determining boundary conditions at these boundaries by potential flow solution. As a result, additional calculations have to be carried out to determine boundary conditions, and the accuracy of computed results also deteriorates. In this paper, the external flow field extended to infinity in both radial and axial directions is

transformed into an internal flow field within a finite region by means of a system of coordinate transformations. By doing so, the free-stream condition and the parabolic flow condition may be used at the outer (infinity in radial direction) and the downstream boundary

(infinity in axial direction), respectively; and the boundary layer flow and the potential flow outside the boundary layer can be deter-mined by an uniform equation system, thus evading this serious

limita-tion. Further steps will be taken to generalize this method to be

applied to the case of ship stern.

In this paper the streamline-iteration method, which has been widely employed for calculating internal flow field in turbine machine

channels, is further generalized to the calculation of turbulent flow around the stern of a body of revolution and its wake using the

popu-lar two-equation turbulence model (K-c) by following our previous works (Jian and Zhou 1981 and Yan et aL, 1982). A curvilinear nonorthogo-nal coordinate system is enployed. One set of coordinate lines

coincide with the mean-flow streamlines, the other set are made of ra-dial straight lines. So the convection terms of the governing

equa-tions for total pressure, turbulent kinetic energy and its dissipation in turbulent flow can be written in the forn cf their variations along streamlines. For static pressure the radial pressure gradient equation

is used. These equations are convenient for numerical calculations.

The mean-flow streamlines, however, are not known beforehand and must be determined by an iterative schome. First, assume the initial

loca-tions of the streamlines and the initial distribuloca-tions of the fluid variables along streamline, then solve the governing equations to

evaluate the new distributions aid the new locations of the

stream-lines. Repeat the calculation procedure until convergence is obtained.

On solving differential equations, assumptions for a thin boundary layer and partially parabolic flow, etc., are exempted.

Calculations have been carried out by using the present method for a body of revolution which is named Afterbody 1 by Huang et al. (1978. The calculated results agree with f-uang's experiment satisfactorily.

For the wake, the accuracy of the computed results by. using the pre-sent method is higher than that of uraoka's (1980a, 1980b).

II. BASIC EQUATION

In calculating the three-d±mnsional incompressible turbulent flow by two-equation turbulence model (K-t), the unknown variables are:

u,v,w ; axial, radial and circumferential

time-averaged velocities in a cylindrical polar coordinate system

p ; time-averaged static pressure

K,c ; turbulent kinetic energy and its dissipation rate

In a cylíndrical polar coordinate system, the time-averaged velocity components and static pressure are governed by following equations:

Continuity equation:

3u

13

16w

-+----

(rv) + O

8z

rr

Axial momentum equation:

3.

13

13

lo lr3

- (uu) +- - (ruy) +- --- (ow) = -- --+- ---- (t )

+

r 6r r

+ - 1--

(rt ) +

(r

rL:r

rz 9

Radial oncntuin equation:

3

13

,1

3

13p

(vu)

+ - ----

_rr

(rvv)

i- -

(vw) - -

= - -

+

roO

r

p:r

+1[-_(

)+f-(rt

)

rz r L,r rr

(t

r-

)1-

_J

r

Circuìferential :.ìntun equacicn:

3 1 1 3 vw

i

3p

- (wu)

+ - -

(rwv)

+ -

(ww)

+ =

-r 3-r r ,9 r r 38 + '

{-(r)

+

r[3r (rT)

+ -

(tr)]

T)

where p is the density;

t

denotes Lie stress, which can be repre-sented by the velocity gradient and che effective viscosity variated

in the flow field. Its tensor form

can

be written as

[T]

eff +

.T)

(5) h e r e

sv

,i

v-

.v

V= ---

e

i -

e

+ ---

e = r r r O 9 :Z Z

The ccmponer.ts of r are:

c'y r = rr err r 1 3w y

eff2r

+ -)]

=u (2)

zz eff 3z 3w w

13v

r =

r.

=

(-z-- - -

+

-rO rr ef f

r

r

r

l3u

3w

t

=

t_

=

.(---)

zO z eft r O 3z 3u c'y

t

=t

=j

(--+---zr

rz

eff ,r

cz) r i

?w,v

i

3u

r

9Tr

r 38

*

In the fornu1as of this

section, the underlined

terns were ne-giecced in the approxizate assunption

of the Spalding's partially

parabolic flcw (see Abde1neuid et al., 1978).

-6-(6)

ore e - equation: -e- (uc) +

4

(rvK) +

4

(wK)

[(4

(r r r r P

Lrtr

_eff,c

13(eff3c)3(eff

_{+CGE_CP)}

2 _f Z

zj

i K 2 K r e ,c e (10

f

'2 ' 2 GE

= u

¿2 1i\ +

(Y

]+f_+i.+

t (,

Uzi

t'ri tr 3 rj

tr

r C C,, C

o-i d eif, etf, 1.44 1.92 0.09 1 1.23

In these equations, and . are ?randtl/Schmidt numbers;

eff,r eff,e

C1, C2 and _CD are proportionality constants. The values of C1, C2,

C , o and : - are given in Table 1.

D eff,K

Table 1. VALUES FOR PR\NDTL/SCHMIDT NUM3ERS AND PROPORTIONALITY CONSTANTS

Equations (1) through (4), (9) and (10) nay be used to obtain the six unknown variables u, y, w, p, K and so they are closure. How-ever, as these equations form a coupled system of non-linear eçuations, it is very difficulty to solve them straightforwardly. Based upon the partially parabolic flow assunption, these equations were solved by using the marching integral technique in the works of Spalding's group. In expressions (3) and (7), is the effective viscosity and de-fined by

'eff = ±

+C

(8)

where , and are the laninar and

turhulent viscosity repc-tively.

The governing equations for turbulent kinetic energy K an ts dissipation rate are:

K - equation: -p-- (uK)

4

-

(rvK)

44

(wK) =

4

r

liff

\

+ r r r p r 'r 0eff,K rJ

+i_ _;'-_

( eff _{_ì}

(llff

::1

+

r2 etf,N- .. eff1r D

Zj

(9)

This paper does -ot adopt the partially parabolic flow ssuption and the streamline-iteration method is used to solve above ecuations.

Cig to

ct

zht the grid is calculated

are situated

cn the

men-rlcw streamlines, these equations can :e rewritten in

following

simple forms.

Continuity equation: '?e employ its intera1 forr

Now if tot-al pressure p

o

-8--

'-.2

-'

erf i , y ,-v 1 cv i

+-I --r---+

-v

r

T rz) y 2 i 2

r

2 w

-(14)

t

_z2

_:t

r

r

_r

U

--eft

flog

"eff

flog

(:LJ

t

-r Z - 4- _ rr r r z w w \w i ? eff I 1 w i 2w 2 v w r2 r2 U

:'r

--+ --

- ---

-

f

l++

r r r r

P t2 :-r2

r r r2 2 1

(flag

eff flog

eff flog 4eff

_t

(15)

T..

+---

.+-r ;r

r

Equations (13) through (15) can be grouped as

( 7)

= -

-

77 +

t!_fL

+ 1C

Jff

[]

(16)

Substitute the formula of vector operation

7)

V () -

J (7 x

into equation (16), then

- - 1 =

V x (7 XV) +

eff +

lOff

[-r])

17)

(18)

If :Y

A _{= Q} = const

(12)

where A is the area of any cross seccica in the channel; Q

is the

mass rate of flow prescribed beforohand. O Enerv equation:

equations (2) through

z

:

r 'O

P

Substitute (4), then

expression (7) and equation (1) into

:Z

rr

2-'

3z2

3r2

r J ¡ 3log ef f i Lllog

eff flog eff

(13)

I

t r

t +

+ T

is introduced and the following relations are utilized

r V(+)

=t°.v()=

along streamline

-o -+

-Z VX

(VxV)=O

where 9. is an unit vector in spatial flow direction, then toflowin equation can be obtained from eçuation (18)

dp d9. Z0 eff

V2'+7 log

eff along streamline (20)

Equation (20),which is called the energy equation of the inccmpress-ible turbulent flow, denotes the change of total pressure of turbulent

flow along the streamline. When _eff o , i.e., the fluid is in-viscid, equation (20) becomes dp /d2 = O . It me.ans

o along stream1ne

that total pressure remains constant along the streamline and becomes the well-known Bernoulli's equation for ideal fluid.

Pressure gradient ecuation:

AfLLr

shifting the terms, equation (16) can be rewritten as

=

-+ eff + V og eff

(21)

Pressure gradients have to satisfy the above equation, which is called the pressure gradient equation. For any spatial curve q , equation (21) can be written in the form of direction derivative aLong curve q

dq q {- ç(V

V)

+ 'eff + log eff

(22)

where ° Is an unit vector in the direction of curve q . Curve q can take three arbitrary linear independent directions. If we take the direction of the streamline to be one of the directions of curve

q , then the energy equation (20) can be obtained from equation (22). For equatIons (9) and (10), if we utilize equation (1) and fol-lowing relation

=(v)

' oz r r 6 di

along streamline along streamline

then the left-hand side of equations (9) and (10) can also be rewritten as direction derivative along the streamline

dKt 1 (1 3 I eff 3K \ 1 3 1 eff 3K

V

----jr

diJ r 3r o _ff,K 3rJ 3eJ

along streamline e r eff,K

+ ' eff + CE -(24) Ql 0eff,K ?z) (19) (23)

i (i 3

( fleff 1

( _'eff a' dLf_{along streamline}

pS%rr

_°eff,c

3r)

r

eff,c

2

di

erf 3ci

+

----j

+ CCE

j - C2p

-j-

(25)

eff,c

For the turbulent flow around the stern of a body of revolution and its wake which is discussed in this paper, the flow is

axisymmet-nc.

Then w -

o ,

and partial derivatives of all flow variables with respect to 9 must be equal to zero. The above basic equations can

be further simplified and we can disscuss the flow case only in a meridian plane. For the axisymmetric turbulent flow, the basic

equa-tions are: Contiruity equation: ri(z)

f

2rr udr ₌ Q = r (z) o

where

r(z)

and r1(z) are the locations of the lower and the upper

(26)

d/dzI

along streamline

denote the direction derivatives of

with

respect to the meridian streamline in and z along the spatial

streamline respectively, the axisymmetric form of equation (20) can be written as

dp

-2

=AhJ

dz

_{along streamline}

uieff t.2

_z

r

r

_r2)

'-2

" CZ Ot

)eff(

u

?r

+

+

(3

(2

)1

(28)

3zJ.J L

z (r

dai

3r

rJJ)

Radial pressure gradient equation: For an axisymmetric flow, we

may discuss the pressure gradient equation (22)

only in the radial

direction.

Thus we have

lo

-boundary of the channel in the meridian plane (z,r)

Energy

equation:

Since

V ₌ V = u

(27)

along streamline along streamline along streamline where V is the meridian velocity, d,/dmI and

n along streamline 2

+

r

-2

rrj

(2

czJ

where GE =

t2[2

+

(2]

± + (29) (

)H

(30) 'eff K eL t ,K

For an axisymmetric

turbulent

flow, the

unknown

variables are u,

v,p,K and

t

. So ecuaticns (26), (8), (2g), (30) and

(31)

are

closure, and can he used to va1uate these unknown variables numerically.

III. cooRDI::ATE TPANSF0tATIONS 0F THE FLOW REGION AND

CALCCL\T1OS

0F THE DERiVATIVES 0F FLOW VARIABLES

The physical flow region of the turbulent flow around the stern of body of revolution and its wake extends to infinity both in the radial and the axial direction (see figure 1). It is impossible to perform finite difference calculation in this infinite region. In

practical numerical calculations, it is a common practice to cut off the infinite region with finite boundaries (dotted lines in figure 1),

resulting a region for performing numberical calculations. However, this practice arouses some difficulties: how to determine the locations arid boundary conditions of the radial cuter boundary and the axial dow-nstream boundary rationally. Additional calculations have to be carried out to determine these boundary conditions (in Maraoka's works the locations of the boundaries were prescribed empirically and the potential flow solutions were taken as the boundary conditions) and the accuracy of computed results also deteriorates. In this paper, we adopt a technique which transforms the infinite flow region into a finite regior with a set of coordinate transformations. As a result,

dr

dz

e. f

:J

z) u 111 J.-_C.L-C

r

3 r Jr C1GE ,

.:\

,

:K r) ±

\

3 3z + i

,--t±.

:z tr

K-equation: a1ong streamline c-equation: along streamline (r ( eff r

:ejf

-

C2o a dv cr CZ 3iong streamline

the free-stream condition and the parabolic flow condition can be em-ploved at the outer and the downstream boundary respectively.

Coor-tra for::aticns re realized y using oiio'ing expressions

r C r

n=

l-e

2

(z

= - arctg(J

Z

where C and C are radial and axial characteristic lengths. C

r z r

is taken as the sum of the radius and the boundary layer thickness at the inlet station, and C is taken as the length of the stern. From

expression (33), it is obvious that for n 0.1 , r = O, and for

= 0,0.5,1 , z

0,C,

, thereby, equations (26) and (28) through

(32) can be solved in the transformed flow region in the coordinate system

(r,) see figure 2.

ifl

this paper, che intersecting of the straight-lines in n-direction (corresponding the radial straight-lines in physical meridian plane) and the transformed meridian streamlines are taken as the grid points. At these grid points, the partial der':atives of flow variables with respect to r and z and the

direction derivatives appeared in equations (28) through (31) can not be evaluated straightforwardly. Some operations must be made.

Let denote a certain flow variable. The first and second

derivatives of :4 with respect to and n along the streamlines

and the station-lines (straight-lines in n-direction),

d/d

d2/d2

,

d'/dn

, d2'/dn2 can be obtained by numerical differential method with the non-equidistant three point difference format (flan,

19S1). Then, by the derivation rule for compound functions, the

direction derivatives of can be obtained in the following form

d' (l-n) dr . . dr C along station-line r d2 d2

l-n2

d (1--\ i -

2c )

dnC j

C dr . dn r r r along station-line

dj 21

27t

=

-

ces ( ) along streamline z d2 r2 1 1 2rd2p dP dz = [; cos

J

Ld2 -tg( )j along streamline

In order to obtain the partial derivatives of flow variables with re-spect to r and z appeared in equations (28) through (31) the

deriv-ative properties and the meridian flow angle n in physical meridian

plane nust 'ra utilized. From figure 3, it is easy to obtain

12

-(34)

---

t,zL+

dz

_{aig streamline}

dr alcng station-line Thus, we h.ve r dr along station-line 2 d

_---tga

3t oz «r along streamline

Perform the derivation operation once again on expression (35), we obtain

(35)

(36)

(38)

On the right-hand side of expression (38),

32/zr and

d2r/dz2 arestill unsolved. with the obtained

P/r

along streamline

and

a2/r2

,

2/az3r

can be derived from the following expression

- -g-

(4k)

()

-

tgci (39)

zr

3z

r

dz ,r 2

along streamline ir

We can first obtain

dId(/r)I

_{along streamline} by nuniberical dif-ferention of

/r ,

and then use the third equation of (34)

tocal-culate d/dz(/r)i

_{along streamline} appeared on the righthand side

of (39). d2r/dz2l_{along streamline} in (38) can be obtained by

deriva-tion of compound funcderiva-tions, employing the coordinates of the grid points in the transformed meridian plane.

C 2[ d2rì

_dn

=

[_

cos2( _)] d

t(-

)J z c 2

l\

₍₄₀₎

along stream line

(dn2(

r + _rj d2 along station-line (37) r2 dr2 32, _d2.

i dr

2 _dz2 along

- -- tga-2

2 streamline r ₂

dz

along streamline d2r dz2

ihus, being given the coordinates n, of the grid points and

now varieble

on these points, we may evaluate its direction deny-tives ¿nd partial derivt ives.

IV. BOUNDARY CONDITIONS

In order to solve the tiow variables u,v,p,K and c from equa-tions (26) and (28) through (31), it is necessary to determine the boundary conditions at each boundary of the flow region. They are defined as follows: ':all Surface

uv=K c=O

Wake Centreline v=O ;

UVdK

_{= o}

r r r òr r

Outer Boundary (infinity in the radial direction)

U=U ,

v=O ;

=cO ; =

o

where y and p are free-stream velocity and static pressure respect ively.

Inlet Boundary

The inlet boundary may be located at about 75% of the length of a body of revolution, where thin boundary layer assumption is valid in general. Thus, the boundary layer thickness can be calculated either by the thin boundary layer theory or by Schlichting's foLunila for flat

plat e

6 = 0.37

( )

where y is kinematic viscosity; S is the distance from the leading

-dø

The velocity components u and y are as follows

1/7 V S

-ÍV

r-r <ô

I 6 j

o-(42)

r-r >6

o

v= O

For static pressure p , either radial uniform distribution and p = p or uniform distribution within and linear variation outside

the boundary layer are assumed.

In order to predict turbulent properties in the flow region pre-cisely, :uraoka improved the inlet conditions for K and c (Muraoka,

1983a). For K , tills paper adopts his proposal, i.e.,

f(r-r ) V2 o

where

f(r-r)

is determined from the experiment dota of Kiabanoff for flat plate (e.g., Rotta, 1972). For the mixing length 9. , we adopt the approximation proposed by Huang et al. (1978).

'(r +6)2-r2

o o

9_

=Q

m mo 3.336

where 9. is determined from the value of g(r-r ) which Bradshaw

no o

et al., have given for thin boundary layer (Bradshaw et al, 1967)

o-g(r-r)

r-r < 1.2 6

mo

g(1.2 ) r-r > 1.2 6

The value of t is determined from the mixing length as follows

3/4

.

[n

(44)

(5) Exit Boundary (infinity in the axial direction)

The parabolic flow conditions are used, i.é.,

=

232v2K

2c - o Z Z 3Z Z

_'

z2 z2 - 2 z2 V. CALCULATiON PROCEDURE

As mentioned above, the present calculation method has to adopt an iterative scheme. The sequence of calculation steps is as follows:

(1) Assumptions of the initial locations of the streamlines

and of the initial distributions for u, K and

t

The calculation is performed in the transformed flow region

which is in the plane. Between interval [0,1] several

straight-lines in ri-direction are taken as calculating station-straight-lines. According

to

a prescribed distribution principle of the mass rate of flow along

r-r < 6

o-r-r > 5 o

the station-line, the locations of streamlines at the inlet station can be dcterminod from the hnon radial distribution of u at the same station by formula (26). Then, keeping the ratios of distances between streamlines at other stations to be the same as at the inlet station, the locations of corresponding streamlines at other stations can be assessed. Join the corresponding points for same streamlines at all stations with smooth curves, the initial calculating grid is formed by the intersecting points of station-lines and streamlines. For other stations, the initial radial distribution for u, K and c

on this grid are assumed to be the same as at the inlet station.

(2) Calculations of the meridian flow angle and of the

distribution for y

According to its definition, the meridian flow angle can be obtained by derivation of compound functions

C

2 r

l-ri

along streamline z

(45)

where

dn/dj

is calculated from the coordinates along streamline

(,ri) of grid points by numerical differention method. Then, y can

be obtained from

V=u

tg

(46)

Calculations of new distributions for K, e and

eff

For each streamline (except on the wall surface and the wake centreline), equations (30) and (31) can be solved simultaneously to obtain the new values of K and e at the grid points on this

stream-line. Firstly, the terms appeared on the right-hand sides of equations (30) and (31) are calculated with the distributions for u, K and e

in previous iteration and the distribution for y in present

itera-tion. Secondly, take the known values of K and e at the inlet

station as the initial values of the integral, and solve equations (30) and (31) by marching íntegration along this streamline from upstream to downstream. The above calculation procedure is repeated for all the streamlines in the flow field. For wall surface and wake

centre-line, K and e

can be obtained with conditions K=c=O and

=

E/r = O

respectively. Thus, new distributions for K and e are obtained. New distribution for is calculated from

ef f

formula (8).

Calculation of the distribution for static pressure p

Solve equation (29) for each station-line to obtain the distribu-tion for p On the right-hand side cf this equation, the distribu-tion for u should take values of the previous iteration and the dis-tributions for y and Pff should take the calculated results at steps (2) and (3) in present iteration. Those terms being known, equation (29) can be solved by ordinary numerical integration. The

16

-dr dn

tga =

interval of integration is from n i to the wall surface or to th.

wake certreline and the integral constant may take boundary condition p = p as its value.

Calculation of thedistribution for total pressure p o

For each streamline (except on the al1 surface and the wake centreline), the energy equation (28) can be solved in the saze way as in step (4). Here, all the terms on the right-hand side of equation

(28) are known, SO _Po can be evaluated by ordinary numerical

integra-tion. The integral constant may take the value of Po at re inlet

station on the corresponding streamline, Po being calculated by

ex-pression (19) using the given radial distributions for u, y and p at the inlet station.

For wall surface, the wall surface condition p p (i.e.,

u=v=O) is used directly. For wake centreline, the xisymmetric con-dition

3u/r = O

utilized to obtain the value of u , then p can be calculated by expression (19) using the obtained u and p and

the axisymrnetric condition v=O

Calculation of the new distribution for

u, p, p

and tg being solved, y and u can be obtained as follows

11=

p

V ₍₄₈₎

/1 + tg2a

(7) Calculation of the new locations of the streamlines.

From the obtained new distribution for u , the mass rates of flow at each station-line can be calculated by the formula (26). According

to the prescribed distribution principle of the mass rate of flow along the station-line, the new locations of the streamlines on station-line,

ocal , can be obtained by inversive

interpolation of the. mass rate of

flow. With a relaxation factor which is less than 1, the assumed new

locations of the streamlines, n , can be obtained as follows.

Re-new

peat steps (2) through (7) until the maximum deriation between the locations of the streamlines in two successive iteration calculations is within prescribed accuracy. Then, these distributions for u, y, p, K and c and the locations of streamlines are the final results

for problem.

VI.

NUMERICAL RESULTS

Numerical calculations are carried out for a body of revolution which is naned Afterbody 1 by Huang et al. (1978). The principal

particulars of this body are given in table 2, and the afterbody con-figuration is show-n in figure 1. Numerical calculations are executed

Table 2. PRINCIPAL PARTICULARS OF AFTERBODY i

L (in) r (in) R

max e

3.066 0.1397

6.6x106

in the transformed flow region which is in the

(-n)

plane. The calculating inlet and exit station are placed at 0.4131 (z/L 0.7553) and = 0.9425 (z/L = il) respectively. The number of the station-lines is 20 along c-direction and the number of the streamlines

is 25 along ri-direction. Station-lines are spaced more closely near the stern and so are the streamlines near the wall surface and the outer boundary (at n = 1) , see figure 2. The calculated results are

shown in figure 1 to figure 6. The calculated mean-flow streamlines in the physical and the transformed flow region are show-a in figure 1 and figure 2 respectively. From figure lit can be seen that in the

region of z/L = 0.90 the mean-flow streamlines are convex in shape and almost parallel to the surface of the body, and the boundary layer is thin. But in the region of the last 10% of the body length the mean--flow streamlines are concave in shape and divergent outwardly and the boundary layer becomes thick. This conclusion is consistent with that obtained by Huang et al. (1978) and Patel et al. (1974) from experiments.

Comparison of calculated and measured velocity components, u/V

and -v/V is shown in figure 4 and comparison of pressure coefficient

C is shown in figure 5. It can be seen that the calculated results by the preseñt method have the saine high accuracy as Muraoka's nunieri-cal results (Muraoka, l980b) both being in fair agreement with experi-ments. The calculated axial velocity profile at z/L = 1.057 by the present method is in better agreement with experiments than Maraoka's.

For turbulent properties, i.e., turbulent kinetic energy K and tnixïng length . , no comparison of calculated and measured results has been given in all the existent papers except Muraoka's (l9BOa). Muraoka improved the inlet conditions for K and and compared calculated and measured results for the first time. The agreement was satisfactory in general, but the accuracy of calculation became worse

as zIL increased. In figure 6 of the present paper, calculated re-sults by the present method are compared with measured

turbulent

-properties by Huang et al. (1978). The agreement etwcun them is satisfactory and the accuracy of calculation at z/L= 1.057 is better than uraoka's.

The calculating exit boundary in uraoka's ca1cultion was located

at z/L=1.182 (1uraoka, 198Gb). As the accurac': of is calculation

became worse, thereby no comparison of calculated and nensured results

at

z/L=1.182

was given, though Huang et al., presented measured re-suits at this station. This paper presents comparisons of calculated and measured results for velocity components, static pressure coeffi-cient and turbulent properties at z/L 1.133 , the agreomnt being satisfactory.

VII. CONCLUDING REMARKS

From the caiculaced example in the previous section, it nay be concluded that:

Based upon the two-equation turbulence model (K-c) , the streamline-iteration method presented in this paper is effective for

the theoretical prediction of the turbulent flow around the stern of a body of revolution and its wake on condition that no separation and

recirculating flow are present. The agreement between neasured and calculated results is encouraging.

The coordinate transformation given in t'ois paper can trans-form the external flow problem which is extended to infinity both in the radial and the axial direction into an internal flow problem in a finite region. Extending the calculating region to infinity in Lhe radial direction, permits us to take the free-stream condition as the boundary condition at the outer boundary and the viscous flow in the boundary layer and the potential flow outside of that can be solved by

an uniform equation system. The serious limitation in the existent differential method, namely the locations of the outer and the down-stream boundary were empirically prescribed and the potential flow solution was taken as the boundary conditions at these boundaries, can be evaded. By extending the calculating region to infinity in the axial direction, not only the parabolic flow condition can be taken as the boundary condition at the exit boundary, but also the accuracy of calculation in wake can be improved.

Works will be continued to extend the present streamline-iteration method to calculations of three-dimensional turbulent flow around ship

stern and of the turbulent flow around the stern of a body of revolu-tion and its wake with propeller in operarevolu-tion.

ACKNOWLEDGMENTS

Before closing this paper, the author wishes to express his heart-ful thanks to Professor Cu Mao-Xiang and Senior Engineer, Liu Xin of China Ship Scientific Research Centre for their advice and encourage-ments. The author is also grateful to Engineer. Wang

Ki-liang of C.S.S.R.C. and to Lecturer. Jinang Jin-liang of Fudan University for their help during the preparation of this paper.

REFERENCES

Abdeleguid, A. 'L, N. C. G. Markatos, D. B. Spalding, and K. Muraoka

(1978). A method of predicting three-dimensional turbulent flows

around ship's hulls, Tnt. Symp.. on Ship Viscous Resistance, SSPA, Goteborg.

Bradshaw, P., D. H. Ferris, and N. P. Atwell (1967). Calculation of boundary-layer development using the turbulent equation, J. Fluid Mech. 20, p. 257.

Harlow, F. H., and p. 1. Nakayama (1968). Transport of turbulence

energy decay rate, Los Alamos Sci. Lab., Univ. California, Rep. LA-3854.

Huang, T. T., N. Santelli, and G. Belt (1978). Stern boundary-layer flow an axisyn'netric bodies, 12th Symposium on Naval Hydrodynamic, Washington, D. C.

flor. Jin-ling (1981). The method of streamline-iteration for

calculat-ing the two-dimensional viscous flow, Shanghai Mechanics 2, No. 1. Jian Jin-ling, and Zhou Lian-di (19S1). Use streamline-iteration

method to calculate viscous flow problems containing both in-ternal and exin-ternal flow field, Shanghai Mechanics, 2, No. 3,

p. 30.

Launder, B. E., and D. B. Spalding (1972). Mathematical Models of

Turbulence, Academic Press, London and New York.

Markatos, N. C., and C. B. Wills (1980). Prediction of viscous flow

around a fully submerged appended body, 13th Symposium on Naval Hydrodynamic, Tokyo.

Muraoks, K. (1979). Calculation of viscous flow around ship ship stern, Transaction of the West-Japan Society of Naval Architects, No. 58, p. 235.

Muracka, K. (1980a). Examination of a 2-equation model of turbulence for calculating the viscous flow around ships, Journal of the Society of Naval Architects of Japan, 147, p. 35.

Muraoka, K. (1980b). Calculation of thick boundary layer and wake of ships by a partially parabolic method, 13th Symposium on Naval Hydrodynamics, Tokyo.

Patel, V. C., A. Nakayama, and R. Damian (1974). Measurements in the thick axisymmetric turbulent boundary layer near the tail of a body of revolution, J. Fluid Mechanics 63, Part 2, p. 345. Rotta, Von. J. C. (1972). Turbulence Strornungen, B. G. Teuber. Yuan fia-le, Zhou Lian-di, and Jian Jin-liang (1982). A

streamLine-iteration for calculating the viscous flow around a axisymrnetric body, Shanghai Lixue 3,. No. 1, p. 57.

-

3.')-1.0

n

1.0

0.8

0.6

0.4

0.2

o

0.4 0.5

S-' - S-.. o Inlet Station 25

21

ft

Streamline 0.75 0.85 0.95 1.05 1.15 1.25 z /L

Figure 1. Afterbody Configuration and Calculated Streamlines for Afterbody i 13 11 9 f i T 0.6 0.7 0.8 0.9 Station Line

Figure 3. Meridian Flow Angle

Exit Station

/

20 t 17 15 >

Fígure

_2.

Transformed Flow Region and Calculated Strearniines for Afterbody i l0 17 16 Grid Fo: -MerIdian Streamline m

-I

4J J)

Velocity Components O

u/V and -v/V for

Afterbody 1

1.0

C G

o - Ca1cu1t ion

:DoAV ::perinent by Huar.g

et 1.

01

3 0 0 -4 3 0

0.2

0.4 0.6 0.8

1.0 1.2

(r-r )/r o max

-

22 -3 z/L=1.C57 .1 S2 - !L-0.7553

z!L0.S-2

0.2 Ou

0.. j.8 1.0 1.2 r-r )/r o max

_r (./

z, L-u.

fl-Z/L0. 942

/L1 .057 - ./L=1.1S2 Calculation

oCOAVO Experiment by ung *t 31.

z/L=0.753

figure 5. Comparison of

-

z/LO.Sô2

Pressure Variations

in the Flow Around

z!LO.9642

.Afterhody 1

z/L1.057

q -

z/L=1.182

o

0.1

O -0.11 -0.05 o 0.1 04 a a a 0.1 0 0 _{o o} z/L=0.344 0.15 -q q

q q

q V

O '-

-

j. q q 0.1 3 0

zJL=0.753

y/LO. 8..62 o

-::/L=0.4

Figure 4. CcrparisOn of

H

/ L0. 9 6 2

'4 p-. 2 _-j

\

z/L"0.7553

'

E

-I

t

.4

. z/L=O.8462

z/L0. 9344

z/LzO. 9642

z/L"l. 182

..a iculat ion

Experiment by Huang et al.

4

0

0.2 0.4

0.6 0.8

0

0.2

0.4 0.6

0.8

(r-r )/r

(r-r )/r

o max o max

Figure b.

Comparison of Turbulent Properties for

erody I.

w

1

xiO3

o cØ*. a _e e e