1

3. Resistance of a Ship3.1 Model testing

• Resistance of a ship:

– Total resistance

• V Speed of the ship

• L Length of the ship

• Density of the fluid

• Kinematic viscosity of the fluid

• g Acceleration of gravity

),,,,( gLVfRT νρ=

ρ

ν

2

3. Resistance of a Ship3.1 Model testing

• Dimensional analysis

– Total resistance coefficient

• Total resistance coefficient

(S = Wetted surface)

• Reynolds number

• Froude number

),( nnTT FRCC =

SV

RC T

T 22/1 ρ=

ν

VLRn =

gL

VFn =

3

• Equal non-dimensional numbers.

- Reynolds number: (same fluid)

- Froude number:

• Conclusion: it is impossible to satisfy simultaneously the

equality of Reynolds and Froude numbers.

• The model dimensions do not allow the equality of the

Reynolds number for model testing.

sm

smnn V

L

LVRR

ms=⇒=

ss

mmnn V

L

LVFF

ms=⇒=

3. Resistance of a Ship3.1 Model testing

Flow similarity

4

• Resistance force is measured at model scale (model) and

extrapolated for full scale (ship).

• Measurements are performed with the equality of the

Froude number at model and full scale (Froude scaling):

• is the scale factor.

• Model’s length is determined by the geometrical

properties of the towing tank.

sm VV 2/1−= α

ms LL /=α

3. Resistance of a Ship3.1 Model testing

Resistance tests

mL

5

• The length of the model should try to minimize the

difference in Reynolds number (maximum length) within

the limits imposed by the towing tank dimensions.

– The precision of the measurements increases with the

growth of the model.

– Model dimensions are limited by the depth (h) and

width (b) of the towing tank section to avoid a

significant influence of the bottom and side walls.

3. Resistance of a Ship3.1 Model testing

Resistance tests

6

• Typical model dimensions:

h: depth

b: width

Area of the model’s midsection < 1/200 bh

– With the reduction of the ship’s length it becomes

difficult to avoid a significant region of laminar flow.

• At full scale, the flow is nearly “fully-turbulent” (region of

laminar flow at the bow is negligible). Therefore, model

testing should avoid laminar flow.

hLm <

2/bLm <

3. Resistance of a Ship3.1 Model testing

Resistance tests

7

• For the typical Reynolds number of model testing (106 to

107), transition to turbulence must be stimulated:

– Trip wires, studs or roughness strips applied at the bow.

These devices introduce an added resistance that has to

be estimated to correct the measured resistance.

– Turbulence of the outer flow may be increased with the

use of grids or bars in the incoming flow.

3. Resistance of a Ship3.1 Model testing

Resistance tests

8

3. Resistance of a Ship3.1 Model testing

Resistance tests

• The model is towed at a

constant speed and it is

generally free to heave,

surge, pitch and roll.

• The resistance force is

measured.

• The test is

performed at

different speeds.

9

3. Resistance of a Ship3.1 Model testing

Resistance tests

10

3. Resistance of a Ship3.1 Model testing

Resistance tests

11

3. Resistance of a Ship3.1 Model testing

Resistance tests

12

3. Resistance of a Ship3.1 Model testing

Resistance tests

13

3. Resistance of a Ship3.1 Model testing

Resistance tests

14

Examples:

http://www.youtube.com/watch?v=Uc35JROubRM&feature=related

http://www.youtube.com/watch?v=MLc-NRKYqis&feature=related

http://www.youtube.com/watch?v=Odkc4ic6jds

http://www.youtube.com/watch?v=RQfzXdTuceY&feature=related

3. Resistance of a Ship3.1 Model testing

Resistance tests

15

3. Resistance of a Ship3.1 Model testing

Resistance tests

16

3. Resistance of a Ship3.1 Model testing

Resistance tests

17

3. Resistance of a Ship3.1 Model testing

Resistance tests

18

3. Resistance of a Ship3.1 Model testing

Resistance tests

19

• Resistance has two contributions:

– Friction resistance.

(Shear-stress at the wall)

– Pressure (residual) resistance.

(Pressure distribution on the ship surface)

• Non-dimensional coefficients:

),()(),( nnRnFnnT FRCRCFRC +=

3. Resistance of a Ship3.1 Model testing

Resistance components

20

• Froude’s Hypothesis:

– The friction resistance may be calculated from the flow

over a flat plate with the same length of the ship

(equality of Reynolds number) and the same wetted

surface. All the rest is residual resistance.

– The residual resistance is independent of the Reynolds

number, i.e. it depends only on the Froude number.

)()(),( nRnFnnT FCRCFRC +=

3. Resistance of a Ship3.1 Model testing

Resistance components. Froude’s hypothesis

21

• Model scale:

– Total resistance coefficient:

– Friction resistance coefficient:

– Residual resistance coefficient:

mmm

TT

SV

RC m

m 22/1 ρ

=

)(mm nFF RCC =

mm FmTR CCC −=

3. Resistance of a Ship3.1 Model testing

Resistance components. Froude’s hypothesis

22

• Ship (full scale):

– Residual resistance coefficient:

– Friction resistance coefficient:

– Total resistance coefficient:

• Correlation allowance,

mms FmTRR CCCC −==

)(ss nFF RCC =

aFsRT cCCCss

++=

ac

3. Resistance of a Ship3.1 Model testing

Resistance components. Froude’s hypothesis

23

• Schoenherr:

• ITTC 1957:

)log(242,0

Fn

F

CRC

×=

210 )2(log

075,0

−=

n

FR

C

3. Resistance of a Ship3.1 Model testing

Extrapolation of the friction resistance

24

3. Resistance of a Ship3.1 Model testingGeosims Lucy Ashton

25

3. Resistance of a Ship3.1 Model testing

Geosims Simon Bolivar

26

CT

nRlog

3. Resistance of a ShipResistance components

Froude’s method

27

C F

C R

C T

log R

F>0

F=0

Resistência

de forma

wFT CCkC ++= )1(

Form (“viscous pressure”) resistance coefficient: kCF

1+k: Form factor

3. Resistance of a ShipResistance components

Viscous pressure resistance “Form drag”

28

CT

nRlog

3. Resistance of a ShipResistance components

Hughes’s method

29

• Wave resistance coefficient

is proportional to

• Total resistance coefficient:

• Therefore,

4nF

4)1( nFT cFCkC ++=

F

n

F

T

C

Fck

C

C4

)1( ++=

F

T

C

C

3. Resistance of a Ship3.1 Model testing

Deteremination of form factor Prohaska’s method

F

n

C

F4

30

• Model scale:

– Total resistance coefficient:

– Friction resistance coefficient. ITTC line:

– Form factor 1+k,viscous resistance coefficient:

– Wave resistance coefficient:

mmm

TT

SV

RC m

m 22/1 ρ

=

)(mm nFF RCC =

mm FmTw CkCC )1( +−=

3. Resistance of a Ship3.1 Model testing

Extrapolation of the Resistance, ITTC method

( )mFV CkC += 1

31

• Ship (full scale):

– Wave resistance coefficient:

– Form factor 1+k independent of Reynolds number.

– Friction resistance coefficient, ITTC line:

– Total resistance coefficient:

• Correlation allowance

ms ww CC =

)(ss nFF RCC =

awFT cCCkCsss

+++= )1(

ac

3. Resistance of a Ship3.1 Model testing

Extrapolation of the Resistance, ITTC method

32

• Roughness effects on the wall shear-stress of a turbulent flow:

– For typical roughness heights smaller than the thickness

of the viscous sub-layer (region with negligible Reynolds

stresses), the wall shear-stress is not affected by the

roughness of the wall, hydrodynamically smooth wall.

sk

ρ

τ

ν

τ

τ

w

s

u

ukk

=

<≡+ 5

3. Resistance of a Ship3.1 Model testing

Roughness effects

33

• Roughness effects on the wall shear-stress of a turbulent flow:

– For typical roughness heights much larger than the

thickness of the viscous sub-layer (region with negligible

Reynolds stresses), the wall shear-stress becomes

independent of the Reynolds number and essentially

determined by the roughness height, fully-rough regime.

sk

8070 −>≡+

ντuk

k s

3. Resistance of a Ship3.1 Model testing

Roughness effects

34

• Roughness effects on the wall shear-stress of a turbulent flow:

– Equivalent sand-grain roughness height, , of a given

surface is the height of an evenly distributed sand-grain

roughned flat plate that produces the same resistance of the

selected surface. This is a single parameter definition of

roughness that is not easy to obtain for real ship surfaces. A

recently painted ship has a typical value of ksg=30µm,

which is equivalent to 150µm for the average roughness

height, kM, (the typical roughness height measured in

shipyards).

sgk

3. Resistance of a Ship3.1 Model testing

Roughness effects

35

• Roughness effects on the wall shear-stress of a turbulent flow:

– The non-dimensional parameter used to quantify roughness

effects is the Reynolds number based on the roughness

height

ν

sg

k

VkR =

3. Resistance of a Ship3.1 Model testing

Roughness effects

36

3. Resistance of a Ship3.1 Model testing

Roughness effects

37

• Roughness effects on the wall shear-stress of a turbulent flow:

– Near-wall non-dimensional roughness parameter depends

on the Reynolds number of the flow

– Model testing is perfomed with “hydrodynamically

smooth” surfaces.

– Full scale ships have rough surfaces.

3. Resistance of a Ship3.1 Model testing

Roughness effects

9.0

1.0

17.02

Ls

Lsfs R

L

k

L

xR

L

kCkuk

−

+

≈==

ντ

38

• Roughness effects are covered by the correlation allowance, .

• The correlation allowance is not only a “roughness correction”.

Each model basin uses its “know-how” to determine .

• Holtrop’s formula for the correlation allowance:

• Bowden and Davison formula:

ac

ac

00205.0)100(006.016,0

−+=−

wla Lc

3. Resistance of a Ship3.1 Model testing

Roughness effects

00064.0105.0

31

−

=

PP

Ma

L

kc

39

• Resistance tests are frequently performed with the rudder

and the remaining appendages (shaft brackets, bilge keels,

fins, etc).

• The appendages contribute to the wetted surface of the ship.

• The Reynolds number based on the ship length (and

undisturbed velocity) is not representative of the local flow

on the appendages. In general, due to the smallest

dimensions of the appendages, the local flow has a smaller

Reynolds number than the ship flow.

3. Resistance of a Ship3.1 Model testingAppendage resistance

40

• The extrapolation of the resistance based on a friction

resistance dependent on the Reynolds number and an equal

form factor at model and full scale, may not be applicable to

each appendage separately.

• Due to the small size of the appendages at model scale, it

may be impossible to avoid laminar flow on the appendages

for the lowest velocity tests, required to determine the form

factor. At full scale, the flow will be “fully-turbulent” on the

appendages.

3. Resistance of a Ship3.1 Model testingAppendage resistance

41

• An alternative way to determine the viscous pressure

resistance of the appendages (“form drag”) is to perform

two model scale tests at high speed for a bare hull and a

fully-appended model. Assuming that the wave resistance is

equal in both models and that the friction resistance

component may be corrected according to the wetted

surface of the two models, the difference between the

resistance of the two tests gives a measure of the viscous

pressure resistance of the appendages.

3. Resistance of a Ship3.1 Model testingAppendage resistance