Hydrodynamic

Hydrodynamic

Introduction

1

Pictures from and based on the books :

Ship Dynamics for Mariners (IC Clark, The Nautical Institute)

Ship resistance & flow (SNAME 2010)

Viscous Fluid Flow (Franck White)

Fluid characteristic

Definition of a fluid : A continuous, amorphous substance whose molecules move freely past one another and that has the tendency to assume the shape of its container; a liquid or gas.

Properties :

• Isotropy : same characteristics whatever the considered directiondirection

• Mobility : it will take the shape of a tank

• Viscosity : is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress

• Compressibility : the density depends on the temperature and the pressure (for water, we consider it’s independent of the pressure

2

Forces on a fluid

• Gravity : volume force

• Pressure : force per surface

• Friction : interaction between particles and surface

• Inertia : proportional to acceleration

• Capillarity• Capillarity

• Surface tension

• Chemical forces

• Magneto hydrodynamic force

3

In general,

smaller than

the other 4.

PressureDefinition

A pressure is :

- a force per unit area

- given in Pascal (Pa – N/m²)

- scalar value (scalar field) meaning independant of direction

n

dS

P

Fp = P.dS.n

Unit Pa is not scaled for pressures induced by water,

in practice kilopascal (1 kPa = 103 Pa) and megapascal (1 MPa = 106 Pa) are used

P

PdS1

PdS2

PdS3

PdS4

PressureDifferent units

• 1 bar = 105 Pa = 0.1 MPa = 100 kPa

• 1 atm = 101,325 Pa = 101.325 kPa = 1.01325 bars

• 1 kgf/cm2 = 9.807 N/cm2 = 9.807 × 104 N/m2• 1 kgf/cm2 = 9.807 N/cm2 = 9.807 × 104 N/m2

= 9.807 × 104 Pa = 0.9807 bar = 0.9679 atm

• 1 atm = 14.696 psi

• 1 kgf/cm2 = 14.223 psi

PressureAbsolute, gage, and vacuum pressures

• Actual pressure at a given point is called the absolute pressure.

• Most pressure-measuring devices are calibrated to read zero

in the atmosphere, and therefore indicate

gage pressure, Pgage = Pabs - Patm

• Pressure below atmospheric pressure are called

vacuum pressure, Pvac=Patm - Pabs

PressureVariation of Pressure with Depth

• Pressure in a fluid at rest is independent of the shape of the

container.

• Pressure is the same at all points on a horizontal plane in a

given fluid.given fluid.

PressureScuba Diving and Hydrostatic Pressure

• Pressure on diver at 100 ft?

100 ft

1

• Danger of emergency ascent?

100 ft

2

If you hold your breath on ascent,

your lung volume would increase by a

factor of 4, which would result in

embolism and/or death.

Statical pressure on a boat

The pressure forces are perpendicular to the plate.

The statical pressure is quite easy to calculate.

9

Dynamic pressure• The static pressure is a kind of

potential energy per unit volume.

• If we make a small hole, because of

this pressure, there will be a jet.

Potential energy will be changed

into kinetic energyinto kinetic energy

• It give the dynamic pressure : ρ g h =

½ ρ V²

10

BernoulliDaniel Bernoulli (Groningen, 8

February 1700 – Basel, 8 March

1782) was a Dutch-Swiss

mathematician and was one of the

many prominent mathematicians

in the Bernoulli family. He is in the Bernoulli family. He is

particularly remembered for his

applications of mathematics to

mechanics, especially fluid

mechanics, and for his pioneering

work in probability and statistics.

(from wikipedia)

11

BernoulliBernoulli’s theorem shows the conservation of energy.

12

It can be written :

ρ g h + ½ ρ V² + p = constant

Bernoulli

Let’s consider this pipe.

Liquid incompressible,

so same volumetric flow

rate : A1V1=A2V2

13

ρ g h1 + ½ ρ V1² + p1 = ρ g h2 + ½ ρ V2² + p2

Because same

pressure

�

1

222

21

1 −∆=AA

hgV

Bernoulli : the pitot tube

The pitot tube measure

the pressure (static

and dynamic) with

one opening and the

static pressure with

the other one.the other one.

14

ρ)(2 st pp

V−=

Bernoulli equationValidity

• The Bernoulli equation is an approximate relation between pressure, velocity, and elevation which is valid in regions of steady and regions of steady and incompressible flow where net frictional forces are negligible

• Equation is useful in flow regions outside of boundary layers and wakes, where the fluid motion is governed by the combined effects of pressure and gravity forces

Bernoulli around a shipAround the hull, the

flow is modified as in

the previous tube.

There are 3 zones (high,

low and high

pressure), � wavepressure), � wave

Two stagnation point :

pressure =1/2 ρ V²

16

Too simple.

No friction considered…

Surface tension

• Due to molecular forces

• Try to reduce the surface for the

volume (that’s why the drops are

spherical).

• In still water, force to « open » the sea• In still water, force to « open » the sea

= force to « close »� no effect.

• In rough water, the spray : it costs

energy

17

Viscosity• Due to intermolecular attractive forces

• When we move the upper plate, there is a resistance force.

• Viscosity is define as the ratio

• So the frictional force : F = η A V / S

SV

AFso

rateStrain

stressShear

/

/=η

18

Viscosity• The classical formulation is (for 2D) :

• Behaviour of the fluids :

• Fortunately, water is a

y

u

∂∂= µτ

newtonian fluid

• Unit of µ Ns/m² or kg/(ms)

19

Laminar flow

All the particle trajectories

are parallel.

The energy is transfered by

the viscosity. the viscosity.

Resistance proportionnal to

the speed of the flow.

20

Turbulent flow

If the speed increases or if the surface length becomes too big, it

will be instable.

�Turbulent

Particles move in all

21

Particles move in all

direction and the

Kinetic energy is directly

transfered.

Resistance proportionnal to the

square of the flow speed.

ViscosityTurbulent flow and boundary layer

• Laminar flow always turns turbulent

• The boundary layer is the region where viscosity must be considered

• Region where energy is dissipated : resistance called form drag

• Out of the boundary layer Bernoulli’s law can be applied

Turbulent flowAt the beginning,

laminar.

After, turbulent.

It occur in the boundarylayer (zone in whichviscosity isviscosity isconsidered).

23

At the end of the plank, the wake.

Bernoulli’s law doesn’t apply as energy is being dissipated in

turbulence. The streamline doesn’t fully converge � increase

of resistance : form drag.

Reynolds

Osborne Reynolds (23 August

1842 – 21 February 1912) was

a prominent innovator in the

understanding of fluid

dynamics. Separately, his

studies of heat transfer studies of heat transfer

between solids and fluids

brought improvements in

boiler and condenser design.

24

Reynolds number

O. Reynolds worked on the

transition of laminar to

turbulent flow in pipe.

He concluded that the transition υη

ρ VDVD ==ReHe concluded that the transition

is function of the ratio inertia

force / viscosity force.

25

υημ is the dynamic viscosity

ν is the kinematic viscosity

Reynolds number for a shipThe length to consider is no more the diameter. We consider the

hull length.

The critical Re (for

transition) is from

0.4x106 to 106 (even

107), function of the

26

107), function of the

hull and the

roughness.

For sea water, μ=1.87x10-3 at 0°C and 0.97x10-3 Ns/m² at 25°C.

So, with μ=1.4x10-3 and ρ=1025 kg/m³, Re=112x107

Transition point is around 0.2 % of the length.

Foil

A flow on a profile produces a

lift and a drag forces.

Great to make the aircraft flying but also for the ships :

27

Great to make the aircraft flying but also for the ships :

Foil

• The force is created by the asymmetrical flow.

• It’s the combination of a symmetrical flow (no lift)

• And the circulation

• Difference of pressure proportionnal to V²

28

Foil

• The drag costs energy and the lift is what we want.

• If the profile is symmetrical and no angle of attack � no lift

• If the profile is asymmetrical or angle of attack � lift

29

Foil• Along the profile, the separation occurs at the end of the

profile.

• So, there is a wake.

• If the angle of attack is to big, the seperation point will be

more in the beginning of the profile � stall

30

Foil• What does the lift depend on?

31

So :

- The angle of the rudder

should be limited.

- The rudder area of a fast

boat will be smaller.

- The force will increase

linearily with the area.

Cavitation• 2 problems : the lifting force can not increase (the difference

of pressure is limited).

• The bubbles appear but collapse when the pressure

decrease� damage

• It can be a problem for propellers

32

Cavitation

• If the difference of pressure is to big, the water will « boil »

(changes state from liquid to water vapour)

• The vapour pressure should counteract the surface tension.

33

Resistance : the separate components

Hull still water resistance

Frictional or ResiduaryFrictional or skin resistance

Form drag

Residuaryresistance

Wavemaking

Eddy making

AppendagesAir

resistance

34

Resistance : skin friction

• Skin friction and residuary resistance are not linked.

• Skin friction is function only of the speed, the viscosity, the

wetted area and the length of the hull.

• So, it depends on Re and the wetted area…

• Tests were done with plates in towing tanks (so no residuary• Tests were done with plates in towing tanks (so no residuary

resistance) and curve fitting has been doen.

35

Resistance : skin friction

• We often work with coefficient of resistance.

• It’s a way to have adimensional value.

SV

RC f

f21 ρ

=

• So, following the ITTC conference of 57 :

36

SV 2

21 ρ

( )210 2Relog

075.0

−=fC

Eddy making resistance• If the change of flow direction is too severe (>20°), it will fail

to follow the contour

�Separation and creation of eddy making resistance

� Increase of resistance and, here, problem for steering

37

Eddy making resistance

Separation occurs later when turbulent boundary layer (water

« fills » more easily the available space)

� For eddy making resistance, it is better to have turbulent flow

It can also appear in the fore part, if the waterline is too convex.It can also appear in the fore part, if the waterline is too convex.

38

Eddy making resistanceWhen a ship is relatively slow moving for its length : 2 main components

of resistance :

- Friction

- Eddy making resistance if bad shape

Spherical to reduce the wetted area

Aft part very narrow to redure eddy making resistance

Large bow because slow speed, so no wave

� Cods head and mackerel tail (1585)

To increase the deadweight, adding

at the midship section39

Lord KelvinWilliam Thomson, 1st Baron Kelvin,

(26 June 1824 – 17 December 1907)

was a mathematical physicist and

engineer.

At the University of Glasgow he did

important work in the mathematical important work in the mathematical

analysis of electricity and

formulation of the first and second

Laws of Thermodynamics, and did

much to unify the emerging

discipline of physics in its modern

form. Lord Kelvin is widely known

for realising that there was a lower

limit to temperature, absolute zero. 40

Kelvin wave pattern

Pattern of waves following a

ship

41

Kelvin wave pattern of a moving

disturbance• Group velocity of a wave is

the velocity with which the overall shape of the wave's amplitudes

• The phase velocity of a • The phase velocity of a wave is the rate at which the phase of the wave propagates in space

• Here, group velocity=0.5 phase velocity

42


disturbance

• Speed of the wave phase Cw =V sin Q

• Speed of the wave group Cg =0.5 V sin Q

43


disturbance : two kinds of waves

• Transverse waves: same phase velocity, perpendicular to the

motion

• Divergent waves : slower phase speeds, angle which

decreases for waves of lower phase speed. (includes a whole

spectrum of waves)spectrum of waves)

44

Kelvin waves : submarine

• During the 2nd world war, the waves created by the periscope

made them visible…

45

Kelvin waves for a shipOn a ship, creation of such wave system on points where we

have change of pressure gradient. On a ship : 2 points

• High pressure centre at about 5% aft of the bow where the

streamlines start to converge causing pressure to reduce

downstream, so the waves originates as crests

• Low pressure at ~5% forward the stern, divergence of

streamline, pressure increases � troughsstreamline, pressure increases � troughs

46

Interference

• Because wavelengths depend on the speed and 2 systems of waves

are created � Interference between the waves

• Speed of wave:π

λ2

gV =

• Half l:

• Number of half l:

• Because 180° difference of phase: odd N : constructive interference

even N : destructive 47

g

V 2

5.0πλ =

2

9.0

5.0

9.0

V

LgLN PPPP

πλ==

Trend in wave making

48

Wave resistance• Fr = 0.38 is the limit for displacement ship

• (Friction resistance has to be added)

• Above that, the bow wave increases.

• To reduce the wave resistance, the waterline should be as

smooth as possible.

• But contradiction with the goal of merchant ship which is to

increase the deadweight � concave shape

• Contradiction with seakeeping performance (concave ships

have more buoyancy reserve). 49

Bulbous bow

• Goal of the bulbous bow: to create a wave, which will makedestructive interference

• Problem : it is done for certain speeds. At different speed, wemay have constructive interference

• Other advantage: add forward buoyancy � waterplane maybe finer

50

Appendage resistance

• Rudder, stabilisers fins, propeller, etc increase the resistance

• Not placed for towing tank test

(too many variable)

• They have their own Fr and Re

51

Air resistance

• In air resistance, we consider frictional and

eddy making resistance

• In calm conditions : ~4%• In calm conditions : ~4%

• When wind, it can increase considerably

52

Form drag

• Frictional resistance is considered equal to the resistance of a

flat plate with the same wetted area

• But, if we make test at low Froude (so wave making resistance• But, if we make test at low Froude (so wave making resistance

can be considered as negligible), the total resistance is not

the frictional resistance. There is an additional residuary

resistance : the form drag

• Form drag is siginificant for wider boat

53

Form drag

• It is due to the boundary layer which is thicker when the

beam to length ratio increase.

• Bernoulli flow is forced to undergo a greater acceleration,

which make the boundary layer thickness.

• The stern pressure is lower, so the wake is bigger. • The stern pressure is lower, so the wake is bigger.

54

Towing tank

• Why?

• CFD is not yet very accurate to estimate the power of a boat.

• Statistical laws are limited

• Is it possible to use the results?• Is it possible to use the results?

• Yes, with some conditions…

55

Towing tank

3 kinds of forces are involved :• Inertial force (ma)

• Gravitationnal force (mg)

proportional to r U² l²

proportional to g l³r• Gravitationnal force (mg)

• Viscous force

56

proportional to µ U l

If the ratio of these forces are

the same, the flow will be

similar

Towing tank

gl

U

lg

lU

Gravity

Inertia 2

3

22

==ρ

ρ

µρ

µρ Ul

Ul

lU

Viscous

Inertia ==22

gl

UFr =⇒

νµρ UlUl ==⇒ Re

57

µµUlViscous

Viscous

Inertia

Gravity

Inertia

Viscous

Gravity1−

=

νµ

So, it means that if the Re and

the Fr numbers are the

same, the flows will be

similar.

Same Fr and Re numbers

λ=M

S

l

l

λS

S

MSM

SMSM

U

gl

glUU

gl

U

gl

UFrFr ==⇒=⇒=

The scale

58

λSSM glglgl

23ReRe

λννν

ννS

S

M

S

MSM

S

SS

M

MMSM l

l

U

UlUlU ==⇒=⇒=

Great, we can have similar flows…

We just need to respect the two relations above.

No problem, let’s replace the water by a liquid with another

viscosity, there is just 2 100 000 l to put and if we change the scale, we will replace it again, it’s easy

Towing tank test

• Following Froude, friction and residuary

coefficient are independent.

( ) ( ) )(ReRe, FrCCFrC RFT +=

• So, if we can obtain the friction resistance, we

can calculate the total resistance with respect

of Froude number.

59

Towing tank test

• Froude’s method :

• Perform the resistance tests with the model.

• So, we have R

( ) ( ) )(ReRe, FrCCFrC RFT +=

• So, we have RTM

• We know that :

• And that for the model and the

ship

60

MMM

TMTM SV

RC

221 ρ=

RFT CCC +=

Towing tank test (2)

• Following ITTC 57 :

(we can calculate it for the model and the ship

• Calculate CRM

• Thanks to Froude similitude :

( )210 2log

075.0

−=

RnCF

FMTMRM CCC −=

• Thanks to Froude similitude :

• We make the same procedure by the other

side…

61

RSRM CC =

Towing tank test (3)

• Following ITTC 57 :

for the ship

• Calculate CTS

(the last term is the roughness allowance :0.0004)

( )210 2log

075.0

−=

S

FSRn

C

FFSRSTS CCCC ∆++=

(the last term is the roughness allowance :0.0004)

• Finally :

• So, we can calculate the power :

62

SSSTSTS SVCR ×××= 2ρ

STSE VRP ×=

Towing tank test (ITTC-78)

• Some differences with the 57th method. The

decomposition is in a viscous resistance, which includes

the form effect on friction and pressure and wave

resistance.

• Assumption is : ( ) ( ) ( ) ( )FCCkFnC ++=+ Re1Re• Assumption is :

• Compute CFM

• Calculate the form factor k

63

MMM

TMTM SV

RC

221 ρ=

( ) ( ) ( ) ( )nwFT FCCkFnC ++=+ Re1Re 0

Towing tank test (ITTC-78)(2)

• Calculate CWM

• Remark : the wave resistance is smaller than the

residuary resistance for Froude method

• Compute the roughness allowance (according to

Bowden): 1 Bowden):

Where kMAA is the roughness in microns according to the

MAA method. ITTC recommend 150 microns.

• Determine the air resistance coefficient:

Where AT is the frontal area of the ship above the waterline64

S

AC T

AA ×= 001.0

33

1

1064.0105 −×

−

×=∆L

kC MAA

f

Towing tank test (ITTC-78)(3)

• Calculate the total resistance coefficient CWM

• Calculate the total resistance coefficient as before.

( ) AAFWSFSTS CCCCkC +∆++×+= 1

• Calculate the effective power as before also.

65

Form factor

• It includes the ratio of the viscous resistance and the

resistance of the equivalent flat plate.

• So, it includes the form effect.

• Empirical formula (Watanabe):

• Another way is to calculate it at low Fr (<0.15) (Cw=0)

but small forces, so problems on measurement

66

T

B

B

L

Ck B

26.25095.0

+−=

Form factor

• Method of Prohaska : assumption: wave resistance

coefficient is proportional to the 4th power of the Fr.

• So:

• Or:

( ) 411 FnkCkC FT ++=

( )T Fnkk

C 4

1 ++=

• If the assumption in the wave resistance is correct:

67

( )FF

T

Ckk

C 11 ++=

Displacement and planing

• The wavelength from the wave pattern

increases with the speed

Picture from Architecture Navale, D. Presles and D. Paulet

Displacement and planing (2)

• When the wave become longer than the ship

length, the ship should pass its own wave �

Planing



Displacement

ship

Force : buoyancy

only

Planing ship

Semi-Planing

ship

Force :

hydrodynamic lift

Force : buoyancy

and hydrodynamic

lift



• To pass from displacement to planing, the ship needs a adequate

hull and the power.

• To calculate the limit speed : easy with the speed of the wave and

is equal to Fr=0.38

• If the hull is not done for, the ship will never planed but the speed • If the hull is not done for, the ship will never planed but the speed

will never increase more than the limit speed

Copyright DN&T


• The resistance curves can be different, but globally, they ressemble

to the following ones:

Displacement

shipPlaning ship

Limit speed

Resistance curve of serie 60

1000

1200

1400

Friction

Residuary

Total

0

200

400

600

800

0 2 4 6 8 10 12 14

Re

sist

an

ce (

kN

)

Speed (m/s)

Planing

• A part of the flow goes forward : spray

• Hard chine is better.

• The weight has to be lower

• Planing hull is common for pleasure

craft (in some case, not enough

buoyancy)

Picture from Ship dynamics for mariners, I.C. Clark, http://www.realitymod.com and wikipedia

The wake• Because the viscosity, the ship carries away

water.

• It is called the wake.

• The wake affects the flow near the propeller,

and it has a consequence on the propeller and it has a consequence on the propeller

efficiency

• The flow is not uniform and the wake depend

on the distance to the hull and the shape.

Pictures from http://www.qm2.org.uk

The wake (2)

• It is expressed as a ratio of the ship speed which is lost in the wake Container carrier

with 2 propeller

Supertanker

with 2 propeller

(with Vs the speed of the ship

and Va the propeller advance

trough the water)

Pictures from Helices marines, Max Aucher

s

as

V

VVw

−=

Cargo with a single screw Super tanker with a

single propeller

Shallow water

• The Bernoulli pressure distribution distorts the waterline.

• It will be more pronounced if the depth is small.

• Between the river bottom and the hull, water is accelerated,

creating a depression � reduction of the under keel

clearance, called the squat.clearance, called the squat.

• It depends on :

– Static pressure, so it will increase in proportion of V²/g

– The sectional area of the water flow (� blockage factor)

– The block coefficient (the flow will be more restricted in case of high

Cb

77

Squat

• Squat is NOT an augmentation of the draft.

• It is the total reduction in under keel clearance.

• (water level also goes down)

78

Squat

• Blockage factor :

• So,

• Squat can be like :

( )015.0 WWD

dBS

+×=

S=Ship’s immersed midship sectional area

Sectional area of the unobstructed canal

• Squat can be like :

79

( ) mB

n CKSKg

VKS 32

2

1 ××=∆

Squat in narrow channels

• Following A. D. Watt :BCS

g

VsSquat ××=∆ 2

2

2.2

• With

• And V the speed in m/s (and g=9.81 m/s²)

80

SC

S

AA

AS

−=2

Squat in narrow channels

• Following Dr C. B. Barrass:

• With

Bk CS

VsSquat ××=∆ 81.0

08.2

20SA

S =• With

• And Vk the speed in knots

• Attention: these formulas are available in a narrow channel

81

S

S

A

AS =

Comparison of the method

• Speed : 8 kts

• Sectional area Ac = 0.5 (40+60) x 12 = 60 m²

• Sectional area As = 8 x 20 m²

• Block coefficient : 0.8

• Following Watt : SSquat 8.0160514.08

2.2 ××××=∆• Following Watt :

• Following Barrass :

82

mSSquat

SSquat

1.1

8.0160600

160

81.9

514.082.2

=∆

×−

×××=∆

mSSquat

SSquat

04.1

8.0600

160

20

881.008.2

=∆

×

×=∆

Squat in open shallow water

• The previous formulas were available for narrow channel, but

in shallow water, the squat phenomenon is also present.

• Dr I. Dand proposed a formula :

Bk CD

dVSSquat ×××=∆ 2

95

1

• With : Vk speed in knots, d deep water draft, D water depth

and CB the block coefficient

83

Bk CD

VSSquat ×××=∆95

Squat in open shallow water

• Barrass proposed an empirical formula. His philosophy was to

consider a width of influence, function of the beam of the

ship.

• Width of influence :

• Open water blockage factor S

• Open water squat

84

)(04.7

85.0 mC

BF

B

B =

Effect of squat on trim and list

• Distorsion of waterline may change the fore and aft position

of the center of buyoancy.

• If a vessel’s centre of buyancy is forward of midship (the bow

is fuller than the stern)� head trimming momentis fuller than the stern)� head trimming moment

• Faster flow on the fore part, so more « succion » � head

trimming moment

• Acceleration on the propeller � stern trimming moment

85

Squat over a shoal

• If the water depth is small: constant squat…

• But if the vessel sails over a shoal ?

86

Squat and heel

• What about the heel?

87

Other effects of squat

• Frictional resistance is increased, wave making resistance also

� the ship slows down

• This increase of resistance loads more the propeller � more

slip and the propeller revolution tend to decrease

• Proximity of the seabed � greater vibration• Proximity of the seabed � greater vibration

• Increase of turbulence and vibration under the stern � if soft

sediment, water can be discoloured.

• Higher bow wave

• Response to helm action slower

• Motions (rolling, pitching) tend to be dampened by the

cushioning effect of the seabed

88

Wave making resistance in shallow

water

• Waves depend on the water depth (when water depth is

reduced to less than ~40% of the wavelength, it’s influenced

by the seabed).

• Phase and group speed decreases

• First, the waves with longer wavelength are modified : higher• First, the waves with longer wavelength are modified : higher

and longer

89

l (deep water)=

l (12m water)=

mg

V64

2 2

=π

mD

g

V101

2tanh

2 2

=

+λππ

Waves in shallow water

• Waves are longer when depth decreases

• So, angle of 19.28° is no more available

• It will increase when the speed increases and the depth

decreases.

90


• Limit speed : kind of wall in front of the ship: as sound wall

• After this limit, resistance decreases

91


• This effect was discovered accidentally in British canals, around

1844 when barges were towed by horses.

• A horse took fright and ran with the barge.

• The prominent bow wave suddenly disappeared and the speed

was much more bigger. was much more bigger.

• It was because :

– Canals were artificially built with a depth around 1 m (critical speed 3 m/s)

– Barges: long and narrow

– Barges towed from ashore, so no squat by the propeller

92

Steering a ship

•To change the boat's course :

A kind of centripetal force is produced to

maintain the boat's circular motion.

Picture from Ship Dynamics for Mariners, I.C. Clark

Circular motion

Let's consider weight attached by a rope, turning along a point C

� Between 2 successive positions, the velocity change of

orientation.

� So there is a acceleration perpendicular to the velocity

� If acceleration --> force� If acceleration --> force

� This is the centrifugal force


Action of a

ship's rudder


Turning circles

� With the delivery of a

ship, some data have to

be provided.

� For example, the turning

circles.

� To provide for different

speeds, wind, draft, seas,

water depths,)

It can be more than double

in very shallow water


Effect on the propeller

• When a ship turn, resistance of the hull and drag from the rudder increase � Ship velocitydecreasesdecreases

• For its rpm, the slip is higher � increase of charge on the propeller

• It can lead to the maximum power of the engine, if the system tries to maintain the rpm.

Transverse thrust

• Because of the wake (flow velocity on the top of the propeller is lower than the flow velocity on the bottom), the flow angle is less effective and may leadto stall of the upper blade, in one turning direction.

• Turning radius may be different on port or starboard


Pivot point

• Giration of the ship is combined with drift.

• The rotation is around a point in front of

centre of gravity called the dynamic pivot

point P2 point P2


Pivot point (2)

• Giration radius from a point from stem to

stern will decrease up to the Dynamic Pivot

Point and increase up to the stern

• An example of value : • An example of value : – Typical full helm turning radius of 2 L

– Dynamic pivot point ≈ 0.35 L

– � Drift angle of G : 10°

• The only point with no drift is Dynamic Pivot

Point

Force and acceleration

• The force at the stern provides to the ship transversal and

rotational acceleration

• Transversal acceleration depends on the mass and the

transversal forces

• Rotational acceleration depends on the moment and the mass • Rotational acceleration depends on the moment and the mass

inertia


Force and acceleration (2)

• Inertia represents the distribution of the mass along the ship,

around the centre of gravity.

• So, lateral acceleration :

...233

222

211

2 +++≈=∑ RMRMRMMRI

]/[ 2smFR• So, lateral acceleration :

• And rotational acceleration :


]/[ 2smM

FR

]/[ 2sradI

FX RR

Static pivot point P1

• Dynamic pivot point P2 occurs when

the ship is in movement.

• It is possible to turn a vessel which is

stationary.stationary.

• How? By giving short burst against the

rudder

• It depends on the moment of the

force and inertia

• Pivoting point P1 is different from the

Dynamic Pivot PointMX

IGP

R

=1

Static pivot point P1 (2)

• If mass is concentrated to the center of gravity : very small GP1 (as in

sailing yacht).

• If Cb is low, it means that weight will be more concentrated near the

center of gravity, and inversely


Static pivot point P1 (3)

• Once the ship starts to turn, there will be resistance fromwater flow, which will limit the rotation speed.

• This speed depends on the immersed hull surface and itsdistribution.

• Stern trim will have an effect of the GP1:• Stern trim will have an effect of the GP1:

– A stern trim means that G move backward

– Inertia change with the square of the distance

– Xr change with the distance

� P1 will move forward

Balance of forces

• Due to the fact that the viscous loss increases

along the length on the hull, the aft part of

the hull is less effective to generate a

difference of pressure, so the hydrodynamic

forces acts through a point A, wich is forward forces acts through a point A, wich is forward

of midship.

• So, it create a moment which assists the

rotation.


Balance of forces (2)

• Because of drift, angle of attack of the rudder decreases. .

• So the turning • So the turning moment will be transferred from the rudder to the main hull force


Directional stability

• A ship is said directionally stable if, once the action with the rudder is finished, the ship will be in a steady condition again.

• If hydrodynamic moment is too small (“A” is too close to “G”), directional stability will be too big close to “G”), directional stability will be too big (helm will be heavy and ship will be less manoeuvrable)

• If the distance between A and G is too big, it will turn easily but may be slower to be steady again. Risk of over-steering (if the over-reaction makes a bigger movement in the other side)

Directional stability (2)

• If the hydrodynamic hull force is far enough to provide the

centripetal force and the turning moment, the ship has a

neutral direction stability, and turn around the Neutral Point

N0. (Let’s remark that the point moves in function of the rate

of turn). of turn).


Directional stability (3)

• There is no more force on the rudder, because the flow has no

angle of attack with the rudder.

• (But there is a needed force to swing the bow)


Directional instability

• Directionally unstable if the centre of hydrodynamic force on

the hull « A » is forward the neutral point N0.

• The turning moment is increased, with hydrodynamic force

and centripetal force not changed.

• It will rotate faster on itself than around the turning circle, as • It will rotate faster on itself than around the turning circle, as

car skidding


Directional instability (2)

• An increasing number of ships is directionaly unstable under

some conditions of trim.

• They can be steered thanks to small alternating rudder

movement.

• It depends on the rate of turn, so some ships may be unstable • It depends on the rate of turn, so some ships may be unstable

at some conditions and then, become stable.


Directional instability (3)

• As for car, it is possible the drive by skidding, it just has to be

taken into account when sailing...

• The trajectory will be different, and should be anticipated in

channel or canal.


Factors affecting directional stability

• Unfortunately, the points A (centre of hydrodynamic force) and N0 (neutral steering point) are not fixed for a vessel.

• N0 depends on the centripetal force relative • N0 depends on the centripetal force relative to the turning moment required of a given rate of turn.

• A depends on the flow condition of the immersed part of the hull and the distribution of surface area.

Factors affecting directional stability :trim

• The trim affects the distribution of lateral

area.


More stable Less stable

Factors affecting directional stability :

block coefficient

• Ships with a very high block coefficient have a

bigger wake, with reduces the effectiveness of

the aft part of the hull in the hydrodynamic

force.force.

• « A » is more foreward than N0 � Unstable at

small rudder angle.


Balance of the rudder

• On the rudder, with the flow�

hydrodynamic force

• The position, the magnitude, the orientation depend on the flow and angle of attackof attack

• In function of the position of the stock, the moment on the stock is different.

• Balance is the ratio area in front of the stock/total area (here : area a/(area a + area b))

• Often around 20%

Types of rudders


Types of rudders (2)

• Unbalanced rudder : no more used in modern ship. High torsion in the stock � stronger steering system needsteering system need

• Spade rudder : balance is possible. Smaller torsion in the stock, but higher bending moment


Types of rudders (3)

• Normal framed rudder : balanced, so smaller torsion. Because of the lower support, less bending moment. But more wetted area and more wetted area and structure.

• Mariner rudder : because of horn, stresses are lower.


How to calculate the rudder area

• In general, following Dave Gerr (Boat

mechanical systems handbook) : 2% of the

lateral area for planing hull and 3 to 4% for

displacement motor boat.displacement motor boat.

• Following Gillmer and Johnson (and Det

Norske Veritas) :

+=2

251100 L

BLdareaMin

High efficiency rudder (1)

• Fish-tail rudder : equipped of a kind of

deflector on the trailing edge. Increase the

lateral force. Allow higher angle (more than

35°)35°)

Picture from Boat mechanical system handbook, DaveGerr


• Rotating cylinder rudder : the rotating cylinder

accelerates the flow � lift. More lateral force



• Articulated rudder : increase the lateral force

at smaller helm angle

Picture from Ship Dynamics for Mariners, I.C. Clark and Boat mechanical system handbook, DaveGerr

Way to improve manoeuvrability

• Vertical axis or cycloidal propeller

• Active rudder

• Auxiliary thrusters

• Twin screw, twin rudder• Twin screw, twin rudder

• Azi-pods

Vertical axis or cycloidal propeller

• Voith-Schneider propeller : vertical blades

turning around themselve and around an axis


Vertical axis or cycloidal propeller

• The force can be oriented in each direction.

Very useful for tug boat, inland ferry, etc

Picture from Voith-Schneider

Active rudder

• During the 70’s

Source : http://pubs.usgs.gov/of/1997/of97-512/htmldocs/ship/pics/rudder.gif and Ship

Dynamics for Mariners, I.C. Clark

Auxiliary thrusters

• 2 types : tunnel thruster and azimutal thruster

• Very easy to manœuvre


Auxiliary thrusters (2)

• Advantage : simpler, so cheaper,

but only lateral thrust

� Advantage : force can be

oriented in all directions � allow

the ship to be powered in case of

main engine failure

Picture from http://www.vethpropulsion.com and

http://commons.wikimedia.org/wiki/File:Pourquoi_pas_bow_thrusters.jpg

Twin screw, twin rudder

• Possible to spin

• Higher effect if the engines are far (for

example, on a catamaran)


Azi-pods

• Electric engine on a pod, which can turn

completely

• Rudder is no more needed

Picture from Ship Dynamics for Mariners, I.C. Clark and http://boards.cruisecritic.com/showthread.php?t=1810532

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Hydrodynamic