Examples of calculation of stern flow field using boundary layer theory approach

3 SEP. 193t

ARCI-HEF

SYr1POSIUM ON

"HYDRODYNAMICS OF SHIP AND OFFSHORE PROPULSION SYSTEMS"

HOVIK OUTSIDE OSLO, MARCH 20. 25., 1977

"EXAMPLES OF CALCULATION OF STERN FLOW FIELD USING BOUNDARY LAYER THEORY APPROACH

By

Ichiro Tanaka, Osaka University Toshio Suzuki, Osaka University Yoji Himeno, University of Osaka Prefecture

SPONSOR: DET NORSKE VE RITAS

Ref.: PAPER 8/6 - SESSION i

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Lab.

_{y. Scheepsbouwkj}

Technische

_Hoeschoo

Deffi

Summary

An attempt is made to predict the flow field at the

stern of ships, based on the three-dimensional boundary

layer theory. The calculated predictions are compared with the experimental results for a tanker model and a

container ship model. The results show that the boundary

layer approach for calculating the stern flow field is

promising, but a second order approximation is necessary

1. Introduction

The estimation of the flow field around the ship stern

is an important subject. It is useful for designing the

ship propellers, for more understanding of the resistance

characteristics, and for, improving ship forms. The current

method for predicting the flow field around the stern of

actual ships is to extrapolate the measured values of the

model to the values of the corresponding ship using some

extrapolation method. There are several methods for this

extrapolation, but more extensive research is necessary

to reach a final conclusion on the reasonable method with

a firm theoretical basis.

The recent development in the three-dimensional

boundary layer gives us the tool to predict the flow field

for a given ship form at an arbitrary Reynolds number, Rn.

It also gives us the possibility to derive the

extrapola-tion law itself.

In this paper, an attempt is made to calculate the

flow field at the ship stern using two integral methods

for calculating the three-dimensional boundary layers.

The results of the calculations are compared with the

experimental results for a tankèr and container ship models.

Discussions are made on the utility and the limitations

2. Methods of Calculation of the Flow Field at the Ship

Stern

In recent years, many methods have been proposed for

the calculation of the three-dimensional turbulent boundary

layers. Most of these methods belong to the so-called integral methods, and there seem to be no big differences

between them. We use here two integral methods for the

calculation. The two methods are: The Himeno and Tanaka's

improved method [1], and the method proposed by Okuno 12].

The outline of both methods are firstly introduced for

convenience.

The Himeno and Tanaka's improved method:

In this method there is no need for the small cross

flow assumption. Coles three-dimensional vector model is

used to represent the velocity distribution, and the local

skin friction is automatically derived from it. The cross

section of the ship form is approximated by the N-parameter

mapping function into a circle (the same as the function

used by von Kerczek) . The boundary layer equations are described firstly by streamline coordinates,

then transformed to the coordinates determined by the

line corresponding to equal angular displacement through

the mapped circles, the frame line, and the normal to the

surface, for convenience to numerical calculation.

The moment of momentum equation is used as the

auxiliary equation. The distribution of the shearing stress

similar to the two-dimensional case. The outer flow

condition is given by the calculation of the potential

flow using a slightly simplified Hess and Smith method.

The results of the boundary layer calculation provide

us the viscous part of the total velocity distribution at

the stern for a first order approximation. Now, if the

potential flow close to the surface has a velocity

dis-tribution which differs greatly from the uniform stream

velocity, it will be necessary to take this potential part

of the total velocity into account. As the method to be

used in such case, we proposed a method drived from the

singular perturbation theory [3]. The conclusion of that

method is The total velocity distribution 'UJ is given

by the vector sum of the viscous part described above ,

and the potential part,

/-

, where l.j/is the value of

the velocity obtained from the potential flow calculations,

and is the value of

V

at the surface of the body; namely,

lt

±

w-VO

(1)

In this report, the change of the potential flow

velocity along the normal to the surface is neglected for

simplicity. This means that the boundary layer solution is assumed to give the total velocity distribution in the

viscous field.

In order to compare the results obtained with the

velocity

iT

by its components, 2Ç.,and '1J along

X

y , and

Z

axes, where

X

axis is fixed the intersection of the central plane and the load water plane of the model,

and is directed aftward, _'7" to the portside, and

Z

downward. We obtain the components ZYy and in the cross sectional

plane by projecting

7t

(obtained along the normal to the surface) onto the plane passing through the root of the

norma i.

The method proppsed by Okuno:

This method makes use of the small cross flow assumption.

The used velocity distribution is a modified Mager model,

which can accommodate cases of reversed cross flow. The

other features of the method are common with other approaches.

It uses the streamline coordinates, and the shearing stress at

the wall is expressed by Ludwieg-Tillmann's formula. As the auxiliary equations, the entrainment equation is used in the

main stream direction and the moment of momentum equation

in the cross flow direction. The shearing stress

disribu-tion in the boundary layer is obtained by the mixture length

theory. The rest of the calculation procedure

in this

method is the same as the Himeno and Tanaka's improved

method.

3. Principal Particulares of the Models used for the

Calculations and the Experiments.

are used. Particulars of the two models are shown in Table 1.

Experiments

The measurements of the velocity field at the stern

of the two models were conducted at Osaka University.

Two five-hole pitot tubes (one is of the spherical type

of 8.15 min diameter for the tanker model and the other is

of the modified NPL-type of 4.5 mm diameter for the

container ship model) were used for the measurements.

The pressures of the pitot tubes were detected using

differential pressure gauges. Several square stations

were traversed for each model while it is towed in the

resistance test condition. The speeds of the models were

1.00 m/s for the tanker model and l.69m/s for the container

ship model. The corresponding Reynolds numbers were 3.08 x 106 and 5.26 x io6 respectively.

Calculations

The calculations were made using the two methods for

each model. The models were assumed to be at full load condition without trim and wave making. There were several

minor differences between the calculation procedure in the

two methods. The number of panels used for the potential

flow was 260 for the tanker model and 252 for the container

ship model, both in one side of the model. As the friction

was used in the Himeno and Tanaka's improved method, while

in the method of Okuno, Prandtl-Schlichting formula was

used. The starting point for the calculation was at S.S.

9 in the former and at 9-i--- in the latter. The spacings

in the steps of the numerical evaluation were L/200 in

both methods. The caiculations were made at Reynolds numbers correspondinq to those of the experiments.

6. Comparison of the Results of Calculation with the

Experiments.

The results of calculations are compared with the

results of measurements at S.S. . As an example of the

comparison, Figs. l-a to l-c show the results for the

tanker and Figs 2-a to 2-c show the results for the

con-tainer ship.

From these figures it is noticed that the agreement

between the results of the two calculation methods and the

experiments is not satisfactory. However, it can be said

that the calculation results are able to represent the general trend of the measurements. For example; the lines

of equi-velocity component in the

X

direction, , where

V,is

the model speed, are predicted to converge near the bilge and to diverge widely toward the load water line.

The vorticity component in the

_X

direction ÓJx which expresses the so-called bilge vortex or sometimes

longi-tudinal vortex, is generated near the bilge and grows upward

by the convection and diffusion in the flow field. Th

pattern of velocity components in

theY

and

2

directions, and , shows a trend similar to the pattern detected

using tuft grid, with a rotating motion of the flow near

the propeller center due to the existence of the bilge

vortex.

On the other hand, the figures show that the prediction

of the velocity distribution at the stern from the turbulent

boundary layer approach seems to have limited accuracy.

it appears that it is difficult to obtain good quantitave

results using a first order approximation like the present

trail.. The main reasons for the discrepancy between the

calculated and measured results may be stated as follows

The assumption of thin boundary layer is not adequate

for the calculations of the boundary layer at the stern.

It may be necessary to treat the thick boundary layer

including the variation of the presssure in the normal

direction. The boundary layer displacement effect on the outer potential flow may have some role, and it may be better

to account for that effect.

The calculations should be extended to include the

effect of wake on the boundary layer calculations.

The negligence of the wave making and the change

of trim and dipping may have some role in the discrepancy.

considered. in order t'o assess the effect of scale on the velocity distribution at thr' ship stern,

we cite here the results of the cooperative tests conduct-ed on geosims of the container model of lengths 2, 7, and IO

meters in addition to the model used in the present study

of 4 meters. This study was made as a part of research work of SR 138 program of the Ship Research Association

of Japan [41 . The results of the experiments are shown in Fig. 3-a excluding the results of the 2 meter model. In

the figure the distribution

of/Çat

the propeller location

2 % L forward of A.!'.) is shown. The scale effect on the measured contour is clear.

Now, to see t,he scale effect on the calculated results,

the calculations by Okuno method was repeated for the

Reynolds numbers corresponding to the geosims lenqths 2m

and 10m. The results of the calculations are shown in Figs. 3-b and 3-c for S.S 0.35 (3.5%L forward of A.P.).

The effect of changing Rn can be noticed, but it is

difficult to get definite quantitative conclusions from

the comparison between the calculated and measured results. Summing up, we can say that the present approach of

using the boundary layer concept to predict the flow field

at the ship stern is useful in giving an approximate

estimation for the general trends of the flow in this

region. For improvinq the accuracy of the predicted values it seems necessary to include the second order quantities.

Acknowledgement

Thanks are due to the Ship Research Association of

Japan and the members of the committee of SR 138 program

References

HIMENO, Y. and TANAKA, I. An Exact Integral Method

for Solving Three-Dimensional Turbulent Boundary

Layer Equation around Ship Hulls. J. of the Kansai

Soc. of Naval Architects, Japan. 159(1975) pp 65-73

(in Japanese). English Translation in Technology Repts.

Osaka University. 26 (1976)

OKUNO, T. Distribution of Wall Shear Stress and Cross

Flow in Three-Dimensional Turbulent Boundary Layer on

Ship Hull. J. of the Soc. of Naval Architects of Japan.

139 (1976) pp 1-12 (in Japanese)

TANAKA, I., HIMENO, Y. and MATSUMOTO, N. Calculation

of Viscous Flow Field around Ship Hull with Special

Reference to Stern Wake Distribution. J. of the Kansai

Soc. of Naval Architects, Japan. 150 (1973) pp 19-26

(in Japanese) . Partly reported in English in Proc.

14th ITTC, Canada. 3 (1975) _{pp 193-202.}

Study on the Improvement of Accuracy in Power Prediction

of High Speed Container Ships. Report of SR 138

Research Program. The Ship Research Association of

List of Tables and Figures

Table 1 Principal Particulars of the Models used for

the Calculations and the Experiments

Fig. l-a Measured Velocity Field at S.s. 1/2 of the

Tanker Model (M 167, 4.5m)

Fig. l-b Calculated Velocity Field at S.S. 1/2 of the

Tanker Model (M 167, 4.5m) by Himeno-Tanaka's

Improved Method

Fig. l-c Calculated Velocity Field at S.S. 1/2 of the

Tanker Model (M 167, 4.5m) by Okuno Method

Fig. 2-a Measured Velocity Field at SS. 1/2 of the

Container Ship Model (M 268, 4m)

Fig. 2-b Calculated Velocity Field at S.S. 1/2 of the

Container Ship Model (M 268, 4m) by

Himeno-Tanaka's improved method

Fig. 2-c Calculated Velocity Field at S.S. 1/2 of the

Container Ship Model (M 268, 4m) by Okuno Method

Fig. 3-a Measured Velocity Field at the Propeller Location

of the Geosims of the Container Ship Model (M 268)

Fig. 3-b Calculated Velocity Field at S.S. 0.35 of the

Container Ship Model (M 268, 2m) by Okuno Method

Fig. 3-c Calculated Velocity Field at S.S. 0.35 of the

Table i Principal Particulars of the Models used for

the Calculations and the Experiments i

Model

Lm

8m

dm

CB M 167 (Tanker) 4.5 0.8182 0.2674 0.802 M 268 (Container) 4.0 0.6154 0.2154 0.572

TANKER MODEL (f1167)

L,5m, S,S.1/2, RN = 3,08x106

EXPERIMENT

Fig. 1-a

Measured Velocity Field at S.S. 1/2 of the

TANKER MODEL (M167)

q,5m

_S,S.1/2

RN =

3.08x106

H-T METHOD

0.3

0.2

0,1

O

Fig. l-b _{Calculated Velocity Field}

t SS. 1/2 of the Tanker Model (M 167, 4.5m) _{by Uinieno-Tanaka'} Improved Method

TANKER MODEL (f1167)

L415rn

_{S.S,1/2, RN}

_3,08x106

OKUNO METHOD

B.L1

Fiq. 1-c

_{Calculated Velocity Field at}

_s.s.

_{1/2 of the}

CONTAINER SHIP MODEL (M268)

¡4m, S,S,1/2,

RN =

5,26x106

EXP ER IMENT

CONTAINER SHIP MODEL (M268)

L4m

S,S,112

RN =

5,26x106

H-T METHOD

po

E. L.

Fig. 2-b Calculated Velocity Field at S.S. 1/2 of the

Container Ship Model (M 268, 4m) by

Himeno-0.1

CONTAINER SHIP MODEL (M268)

4m) S,S.1/21 RN

=

5,26x106

OKUNO METHOD

0.1

_o o

-0.1

-0.2

m

B.L.

CONTAINER SHIP MODELS L4m, _{7m) 10m}

RN = 5.3x106, 1,3x107., 2,3x107

EXP ER I MENTS

0.05

' o

B.L.

Fig. 3-a _{Measured Velocity Field at the Propeller Location}

o

0.1

CONTAINER SHIP MODEL

2m) S.S. 0.35

RN = 2,5x106

OKUNO METHOD

01

t I VM

vy)

vx/VM

I

O

B,L,

Fig. 3-b

Calculated Velocity Field al S.S. 0.35 of

the

0.2

m

o

CONTAINER SHIP MODEL

10m,

S,S1 0,35, RN =

2,5x107

OKUNO METHOD -)XL/VM

0.1

0.2 ni

B.L.