Further investigations on pressure variations around cylinders of cross section similar to ship water lines

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UDGIVET AF DANSK INGENI0RFORENING

FURTHER INVESTIGATIONS ON

PRESSURE VARIATIONS AROUND

CYLINDERS OF CROSS SECTION

SIMILAR TO SHIP WATER LINES

By Sv. Aa. Harváid.

Professor, dr. techn., Shipbuilding Dept., The Technical University of Denmark

Summary

Calculation of pressure distribution in the potential

field around cylinders of cross section similar to

ship water lines has been performed. The calculations have been made for cylinders placed at certain distan-cés from,a wall. The influènce ofthedistance between the cylinder and the wall has been examined. Laplace 's

equation of continuity for incompressible fluids has

been employed in the calculations, ignoring the

influ-ence of friction. The calculations have been carried

out on a FACIT EDB-computer using a standard

pro-gramme No. B 150. A system with about 150Q unknowns

has been used. Introduction

Navigating along a shore and when sailing on rivers and canals, a heavy suction may occur between ship and shore or bank. In such cases the water depth will

be greatly limited and the flòw will therefore be nearly

two-dimensional. By calculating the pressure distri-bution around cylinders with cross sections as water-lines it is possible to form anideaof the forces acting

on such a body. Calculations of pressure distributions in a two-dimensional flow have previously been carried

out, fôr instance by D.W.Taylor in "On Ship-Shaped

Stream Forms" published in 1894 1 and by the

author in "Pressure VariatiOn at Cylinders of Cross

Section Similar to Ship Water Lines", 1962 J 2

For these investigations the cylinder I-B-2 is se-lected as a ship. The length of the water line is

spe-cified as 100 units, thé breadth as 15 units, the fullness

6 (= cx) as 0.83, and the following seven cases have

been calculated:

A ship sailing in an unlimited large area (indicated by )

A ship sailing parallel to a quay at a distance of 0.3 L (= the distance between the centre line of

the ship and the quay. L is the length of the ship

waterline) (indicated by 30) as 2) but at a distance of 0.2 L (indicated by 20)

as 2) but at a distance of 0.1 L

(indicated by 10)

A ship meeting a current at an angle of about

11.5 degrees. Unlimited large area

(indicated by /)

as 5) but with a quay inserted parallel to the

direction of the current. The distances between

the stems and the quay are 0.2 L and 0.4 L res-pectively (indicated by 20/40)

as 6) but with distances of Od L and 0.3 L (indicated by 10/30).

FURTHER INVESTIGATIONS ON PRESSURE VARIATIONS AROUND

CYLINDERS OF CROSS SECTION

SIMILAR TO SHIP WAtER LINES

By 5v. Aa. Harvald. Professor, dr. techn., Shipbuilding Dept.,

The Technical University of Denmark

Mathematical Basis

Laplace's equation of continuity for incompressible

fluids for a two-dimensional flow becomes in ordinary C artesian coordinates

x2 òy

where denotes the velocity potential function. The X- and Y-component velocities become

u =-2

If the area in which the flow takes place is split up

into a number of elements, the quantity of the fluid

passing through an element is determined by

Q = X - ç, (3)

where X = the conductivity of an element (assuming the value of O or 1)

a = the sectional area of the element (in the

two-dimensionalfield = the breadth of the element)

1 = the length of the element

= the potential difference between the ends

of the element.

The area is split up in two directions normal to

each other.

Assuming a

XT = p

for a point in any element

NN + +

+ WW

-

is obtained

and, for example, code No. B 150 from the

programme-library of the FACIT. Electronics AB may be used.

This programme has been written by Yngve Palm and

SvenÖberg ¡.3 _J.

The field of flow is assumed to be replaced by a

wire net with rectangular meshes. The, flow through

each wire of the net is one-dimensional and is depend-ent on the dimensions and the conductivity of the wire and on the potential difference between the end points of the wire (see equation (3)).

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Figure 2: Net used for computation of the velocity potential (cylinder I-B-2, posItion 20/40)

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.WZ. /,4fl/ff,y,y,,flfl,Vfl,fl7fl,y/flflfl,Vflflffl,, 17 ,9Z?W7fl'flfl/'Wf)'Zf tZftfl,,Zf//f/ftf,Vfl tftffltffI 11ff '17ff fflfffff fIt 50 lOO ISO 200 250 300 -350 400 450 500 550 000 050 700 750 900 050 900 950 1Q90

Method of Calculation

When, solving the present problem the procedUre is as

follows:

The area of flow must be bounded and thèn split into

strips at right angles to each other. In the middle of. each strip awire is inserted. In this way a net of

rect-angles is produced in which the flow cànbe determined. The wire constants are calculated forall wire elements

by means of equation (3). After this a value of the

velocity potential function is chosen for every point

and the boundary conditions are inserted. The calcu-lations then proceed using a Relaxation Method as the

selected potential distribution is improvedby calcu-lating ç,p at each of the points by means of equation

(5). Repeated corrections to the potential values ("PÄ

to be replaced by 'pB in the whole grid are redUced

successively and the calculations are finished when an

adequate agreement has been attained. A faster

con-vergence can be obtained by replacing

PA by °PA + w(ÇO

-Here is the relaxation factor which is of decisive

importance to the calculation time. The optimum

rélaxation factor is between i and 2 and will, among other things, depend on the shape of the calculation

area.

Assumptions

As the capacity of the computer is limited, and as,

from economical reasons, the number of man- and

machinè hours used for the computations are also

limited, assumptions regarding the field of flow have to be made.

In this case the assumptions are:

The curves for the constant at a certain distance ahead of the ship (the cylinder) and at a distance

astern of the ship are straight lines, at right

angles to the direction of flow (ç,=O and q =1000

respectively).

The velocity potential at a certain distance from the ship andina direction parallel to the direction of motion (or flow direction) varies linearily,

The shape of the ship (the cross section of the cylinder) can be defined sufficiently exactly by means of a limited number of non-conducting wires.

Facit Standard Programme No. B 150

This programme makes it possible to use a rectangular net of about 1.600 rectangles and with4l wires in each

of the two axis-directions. Furthermore it is possible to vary the tightness of the net of rectangles across

the field. The net is made tightest where the reatest

variations of are expected to take place.

The boundary values are stated such that advantage

has been taken of possible symmetry in the field. Examples of the non-uniform rectangular grid are

shown in fig. 1 and fig. 2. In the same figures black

dots indicate the end points of the non-conducting

wires defining the shape of the ship. The conductivity A. is equal to 1 for all wires situated in the liquid area

while it is equal toOfor thewires crossing the surface

of the wall, crossing the surface of the cylinder or

lying inside the cylinder. When fixing the entry values

for , calculations have assumed the -curves are

straight lines and a linear variation of from O to

1000.

In many of the calculations the computer time has

been very long, about one hour.

Method of Dealing With the Data From the Computer

The computed potential valúes have been entered in

the net drawings at their respective places, and on

this basis the equipotential lines have been drawn.

Then the stream lines have been cOnstructed as lines forming right angles with the potential lines (see fig.

3). As it has been necessary to use a wide-mesh net

of only 41 x 41 wires and to stop the computation

be-fore having obtained the required accuracy, adjust-ments to both the potential lines and the stream lines

have been necessary.

Thé velocity along the surface of the cylinder has

been determined by

v=

os

where s is the length of a surface element of the

cy-linder. A curve of as a function of s along the

sur-face of the cylinder has been drawn and y has been

determined by graphic differentiation. Bernoulli's equation gives:

p - p0 =

2

PPo

()2

_()

Where

p = the pressure at the surface of the cylinder

Po = the pressure at an infinite distance from

the cylinder

p = the mass density

y0 the velocity of the flow (or the velocity of

"the ship")

After this, curves of the pressure variations have been drawn. Owing to the uncertainty of the graphic differentiation it has also been necessary here to fair the curves.

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stream lines (I-B-2-co)

Figure 4: Potential lines and

Figure 6: Potential lines and stream lines (I-B-2-20).

Figuré 7: Poténtial lines and stream lines (I-B-2-10)

6

Results

Stream line patterns for a ship sailing in a Unlimited area and for a ship sailing at a distance of 30, 20 and

lo from a quay parallel to the threcticn of motion are.

shown in figures 4, 5, 6 and 7 respectively. The

cor-responding pressure variations are shoi in figure

iL In the same figure a curve has been..inserted

cal-culated using the more exact sources- and sitilcs

method 2 . There is an apprôximate similarity

between the . pressure distribution determined, by the net method and the one determined by the source-sink

method. The deviations aré due to the assumptions which had to be made in order to carry out the

cal-culation's\ using the net method. Closer accuracy

might have been obtained if code No B 150 had been used for determination of the boundary values for the

area employed, by first determining the flow in an

area 10 or maybe 100 times its size.

From fig. 11 it is seen that the presence of the quay

alters the pressure distribution at the side turning

towards, as well as at the side turning away fròm, the

quay If the distance is great between ship and quay

the difference between the pressures will beslight,

but if the distance is small, for instance o the same magnitude as the breadth of the ship, the differences

in pressüre will be very great and the ship will be

sucked against the quay.

Streamline patterns for a ship ssilingmn an unlimited

area and for a ship sailing at two differentdistances from a quay have been shown in figures 8, 9 and 10.

Iii all three cases the longitudinal axis of the ship

makes an angle of about 11.5°. with the direction of

motion. There are only small differences .in the pressuré distributions for the three cases (fig. 12).

Only in the case of least distance to the quay s there a very large under pressure in a small area near the

stem. In all cases there is a ver great difference in the pressure distributions between the two sides of

the ship. This causes a heavy turning moment On the

ship trying to put it

lying across the direction of motion

Finally it has to be noted that the forces produced by rudder and bow thruster are small compared with

the forces arising from the pressures around the ship.

References

i j Taylor, D.W.: "On Ship-Shaped Stream Forms',

TINA 1894, p. 385.

¡ 2 I Harvald, Sv. Aa.: "Pressure Variation at

Cylin-ders of Cross Section Similar to Ship Water

Lines", IngeniØren, Vol. 6, Nô. 3, 1962.

3 j Palm, Yngve och Sven Öberg: "Berältuing av

tvâdimensionell stationär värmeströmnirig med hälp av elektronisk datamaskin. (Estimation of

static two dimensional heat flow with electronic

computer)TT,_{Byggmästaren nr. 2, 1961.}

550 600 630 700 750 807 85 300 350 p000

500 550 600 650 700 750 - 600 650

Figure 8: Potential lines and stream lines(I-B-2-/)

Figure 9: Potential lines and stream lines (I-B-2-20/40)

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Figure 10: Potential lines and stream lines (I-B-2-10/30)

Figure 11: Pressure distributions along cylinders Figure 12: Pressure distributions along cylindes (centre line parallel to the direction of (centre line nialdng an angle of 11.5° with

motion) the motion direction)

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