Factors Affecting Ship Resistance

FACTORS AFFECTING SHIP

RESISTANCE

October 1, 2009 1Dr. Adel Banawan

Ship Hydrodynamics-1

Dimensional Analysis

October 1, 2009 2Dr. Adel Banawan


Above waterpart

Under water part

Ship Speed V

Water Density r

Water Viscosity m

Pressure p

Ship Size (L)

Total ship resistance R could depend upon the

following:

• Speed V

• Size of body, which may be represented by the linear

dimension L

• Mass density of the fluid ρ

• Viscosity of the fluid μ

• Acceleration due to gravity g

• Pressure in the fluid P

October 1, 2009Dr. Adel Banawan


Applying dimensional analysis and assuming that the

resistance R can now be written in terms of unknown

powers of these variables:

(1)



fedcba pgLVR mr

Introducing dimensional quantities into equation (1), we

have

(2)



fed

c

ba

LT

M

T

L

LT

ML

T

L

L

M

T

ML

2232

Hence

(3)



fedbTime

fedcbaLength

fdaMass

222:

31:

1:

So, Solve for a,b, and c in terms of d, e, and f

(4)



edc

fedb

fda

2

222

1

Then from equation (1)

(5)

(6)

where the left-hand side of the equation is a non-dimensional resistance coefficient.



fed

V

p

V

gLVLfLVR

22

22

rm

rr

222,,

21 V

p

V

gLVLf

SV

R

rr

Equation (6) states that

if all the parameters on the right hand side have the

same values for two geometrically similar but different

sized bodies, the flow pattern will be the same for each.



If we ignored for now the third term, then

It is obvious that the first term which is related to the

viscosity is also related to the frictional resistance, while

the second term is a function of g is related to the

residuary resistance.



2

2 ,2

1V

gLVLfSVR

T

r

• The frictional resistance is governed almost

exclusively by viscous effect.

• The residuary resistance is concerned with the

dynamic movement of large masses of the fluid.



Case (I)

Consider a non-viscous liquid in which there is no

frictional or other viscous drag,

And for two geometrically similar bodies (Ship and

model):



2

2

21

V

gLfSVRR RT r



Water Density rs

Water Viscosity

ms=0

Pressure ps

Ls

Speed Vs

SHIP

Water Density rm

Water Viscosity mm=0

Pressure pmL

m

Speed

Vm

MODE

L

Geomtrical Similarity:



31

m

s

m

s

m

s

m

s

m

s

S

S

D

D

B

B

L

L

The residuary resistance of the ship RRs and of model

RRm will be in the ratio



2

2

2

2

21

21

m

mmmm

s

ssss

Rm

Rs

V

gLfVS

V

gLfVS

R

R

r

r

If the value of the function argument is the same:

i.e.



22

m

m

s

s

V

gL

V

gL

m

m

s

s

L

V

L

V22

Then

Corresponding speeds



m

m

s

s

L

V

L

V

Then

where ∆s and ∆m are the displacements of ship and

model respectively.



m

s

m

s

m

s

m

s

m

s

mmm

sss

mmm

sss

smm

sss

Rm

Rs

L

L

LL

LL

VS

VS

VS

VS

R

R

r

r

r

r

r

r

r

r

r

r

3

2

2

2

2

2

2

21

21

i.e., the residuary resistance per unit of displacement is

the same for model and ship.



m

Rm

s

RsRR

Froude’s Law:

When the speed of a ship and her corresponding model

are in the ratio of the square routs of their lengths, then

the residuary resistance varies as the displacement



Case (II)

Consider a deeply submerged body, where there is no

wave making, the total resistance will only be

frictional.



r

VLfSVRR FT

2

21



Ship

Vs

LS

Model

Lm

Vm



s

sssssFs

LVfVSR

r 2

21

m

mmmmmFm

LVfVSR

r 2

21

If the parameter is the same for ship and model. In

other words

and assume νs = νm and ρs=ρm then



m

mm

s

ss LVLV



FmFs RR

1

21

21

22

22

2

2

2

2

sm

ms

mm

ss

mmm

sss

Fm

Fs

LL

LL

VS

VS

VS

VS

R

R

r

r

Rayleigh’s law:

• When the speeds of a ship and her model are

inversely proportional to their respective lengths, then

the frictional resistance of the model is equal to that

of the ship provided that ν is the same.



Resistance Coefficients

Divide all through by 0.5ρSV2



RFT RRR

Wetted Surface Area S

= +



222 5.05.05.0 SV

R

SV

R

SV

R RFT

rrr

RFT CCC

tCoefficien

ResistTotal

tCoefficien

ResistFrictional

tCoefficien

ResistResiduary

Non dimensional Groups

1. Froude Number

William Froude (1810-1871)

V in knots and L in feet



FrgL

V

gL

V

V

gL

2

2

L

VRatioLengthSpeed k

2. Reynolds Number

Osborne Reynolds (1842-1916)

V in m/s

L in m

ν in m2/s



RnVLVL

m

r

Froude’s Law, or Rayleigh’s Law?

Assuming ρm=ρs and νm=νs

1.From Rayleigh’ s codition

Since Lm < Ls then Vm >>>>Vs

Example

If Ls=125 m , Vs=25 knots, and Lm=5 then Vm= 625knots. !!!!



mmss LVLV

m

ssm

L

LVV

2. From Froude’s condition

Since Lm < Ls then Vm <<<< Vs

Example

If Ls=125 m , Vs=25 knots, and Lm=5 then Vm= 5 knots.



s

s

m

m

L

V

L

V

s

msm

L

LVV

Conclusions:

1- it is impossible to satisfy both laws at the same time

2- it is possible to satisfy Froude’s law

3- it is uneconomical and impracticable to satisfy

Rayleigh’s law



Documents

Factors Affecting Ship Resistance