Inviscid free-surface flow computations for a series-60 in a channel

Inviscjd Free-Surface

Flow Computations for a Series-60 in a

Channel

Volker Bertram, Hironori Yasukawa

Consider a ship movingsteadily ahead in the center of a channel of constant depth and width.

We lìmit ourselves to subcritical depth Froudenumbers (F,,JL = U/'

< 1, where U is the

ship speed, g gravitational acceleration and h the channel

depth). In this case the flow about

the ship is steady. The flow is assumed to be symmetrical with respect to the hull center

plane coinciding with the center plane of the channel. The flow is strongly influenced by the

shallow water. Theinfluence of the channel walls is for usual channel geometries of secondary importance. Resistance and squat increase drastically near the critical depth Froude

number. These global changes reflect of changes in local flow details such as the wave

pattern or the pressuredistribution on the hull.

We assume an ideal fluid. Then continuity gives Laplace's equation which holds

in the whole

fluid domain. A unique description ofthe problem requires further conditions on

all boundaries of the fluid resp. the modelled fluid domain:

- Hull

condition: Water does not penetrate the ship's surface. - Kinematic condition: Water does not penetrate the water surface.

-

Dynamic condition: There is atmospheric pressure at the water

surface.

-

Radiation condition (for subcritical speeds): Waves created

by the ship do not propagate

ahead

- Decay condition: Far awayfrom the ship, the flow is undisturbed.

-

Open-boundary condition: Waves generated by the ship pass

unreflected any artificial

boundary of the computational domain.

-

Equilibrium: The ship is in equilibrium,i.e. trim and sinkage are changed such that the

dynamic vertical force and the trim moment arecounter-acted. - Bottom condition: No water flows through the channel bottom.

-

Side-wall condition: No water flows through the side walls.

The hull and the channel side-wall are discretized by higher-order panels (parabolic in

shape,

linear in source distribution) similar to Hess (1979,1981). Above the free surface

point source

clusters as described in Bertram (1990) are located. Mirror images of the elements at the channel bottom enforce that no water flows through the channel bottom.

Similarly, mirror

images at thevertical symmetryplane are used. Free-surface and hull boundary conditions are fulfilled by collocation. The free-surface boundary condition is met in an iterative scheme that lineariz differences from arbitrary approximations of the potential and

the wave elevation, see

e.g. Jensen et ai. (1989). The radiation and open-boundary conditions are

enforced by shifting sources versus collocation points onthe free surface.

The wave resistance problem features two special problems

requiring an iterative solution

ap-proach:

A nonlinear boundary condition appears on the free surface.

The boundaries ofwater (waves) and ship (trim and sinkage) are not a priori

known.

TECHNISCHE UNIVEfi$J1J-i 'aboratoflum voor

-The iteration starts with approximating

- the unknown wave elevation by a flat surface,

- the unknown potential by the potential of uniform parallel flow, - the unknown position of the ship by the position of the ship at rest.

In each iterative step, wave elevation, potential, and position are updated yielding successively better approximations for the solution of the nonlinear problem, Fig.1.

Input; initialize flow

field with uniform flow

N

Set up system of equations for unknown source strengths

Solve system of equations Compute velocity etc. (up to 2.

derivatives) on free surface

Compute new wave height Compute velocity on hull

Compute velocity etc.

on new free surface

Pressure integration New sinkage and trim

-Iteration end?

(STOP

Fig.1: Flow chart of iterative solution

The method was applied to a Series-60 ship (CB = 0.6) and compared to experimental data of the Duisburg Model Basin (VBD). We computed the flow for F = 0.15 for hIT = 3.2 ('deep water') and 1.5 ('shallow water'), Fig.2. T is the ship's draft. 503 elements discretized the hull

up to a height of 0.23T above the CWL. The free-surface grid extended 0.8L in lateral direction (to the channel wall), 0.6L ahead of FP and 0.5L behind AP. 96 . 19 = 1824 elements were used to discretize this area. This discretization resolves the wave pattern somewhat coarsely but is deemed sufficient to capture effects relevant for squat and pressure at channel bottom. 962 = 192 elements were used to discretize the channel wall for hIT = 1.5, 96.3 = 288 elements for h/T = 3.2.

h/T = 3.2 'deep water' i h/T = 1.5 'shallow water' Compute pressure on bottom

under-relax unknown source strengths

Fig.2: Computed cases for Series-60 at F = 0.15

Fig.3: Wave pattern

for F = 0.15; h/T =

1.5 (bottom) and h/T = 3.2 (top)

=

0.025U2/g

Local flow details like wavepattern, Fig.3, pressure on hull, and wave profile

showed no

irreg-ularities and a grid variation (longitudinal shift of the upstream and downstream end

by half

grid-spacing) on the free-surface gave almost exactly the same results. Table I gives the com-puted and measured squat. The nonlinear approach captures the squat well. Measurements of

various towing tanks, ITTC (1993), show considerable scatter of measured squat for

h/T = 1.5,

ranging from T/L =

0.00236 to 0.00326. Our computed resultof 0.00264 lies well within this

bandwidth. The difference of 7% between measured (VBD) and computed squat may be due to instationary flow effects in the experiments as original pressure

measurement data had a

tendency to converge to negative pressures far behind the model. For

comparison, results of a linear wave resistance code, Yasukawa (1989), is given.

Table I Sinkage and trim (positive for bow immersion) for Series 60 at F = 0.15

0.0

-0.1-II»

cP * .! x1L -1.0 -0.8J I -0.6J I -0.4I J -0.2I 0.0F 0.2 0.4 0.6 0.8 1.0

Fig.4: Pressure coefficient Cp along center line on channel bottom; h/T = 3.206; experiments (.), present nonlinear panel code (s),linear panel code

(o)

i

3

. I.oao. .

experiment nonlinear method linear method

h/T

AT/L û

T/L

û

T/L

O

3.2 0.00118 0.00050 0.00119 0.00042 0.00104 0.00056

The pressure on the channel bottom is captured well for 'deep water' by both linear and non-linear method, Fig.4. For 'shallow water', the nonnon-linear code improves predictions including the tendency with a local maximum amidships, Fig.5. The improvements are due to the

con-sideration of squat and local wave trough. The local pressure maximum aft (at x/L = 0.5)

is somewhat overpredicted due to the neglect of viscosity. The error in predicting the pressure minimum is 7%. This difference is attributed to the underpredicted squat.

cP

A

0.1 0.0 'D -0.1 -0.2 -0.3 -0.4 -1.0 -0.8 -0.6 -0.4

0

o 0 0. .0 s 0 0

0Ø0 1

IS

s

s o s 4 1.0

x/L

Fig.5: Pressure coefficient Cp along center line on channel bottom; h/T = 1.5; experiments (.), present nonlinear panel code (.), linear panel code (o)

Acknowledgement

The research was performed during a stay of V. Bertram as a visiting scientist of MHI R&D Center in Nagasaki sponsored by the German Research Association (DFG). The authors are grateful for the assistance of Dr. Müller and Dipl.-Ing. Gronarz (VBD) for updated data on

measurements. References

BERTRAM, V. (1990), Fulfilling open-boundary and radiation condition in free-surface problems using Rankine sources, Ship Technology Research 37/2

HESS, J.L. (1979), A higher order panel method for three-dimensional potential flow,

NADC-77166-30

HESS, J.L. (1981), An improved higher order panel method for three-dimensional lifting potential flow, NAD C-79277-60

ITTC (1993), Report on cooperative experimental program in shallow water, Resistance and Flow Committe, 20. mt. Towing Tank Conf., San Francisco

JENSEN, G.; BERTRAM, V.; SÖDING, H. (1989), Ship wave-resistance computations, 5. mt.

Conf. Num. Ship Hydrodyn., Hiroshima

YASUKAWA, H. (1989), Calculation of the free surface flow around a ship in shallow water by Rarikine source method, 5. mt. Conf. Num. Ship Hydrodyn., Hiroshima