propulsion engineering-02-resistance of ships

Piri Reis University

Faculty of Maritime – Department of Marine Engineering

SM415-Propulsion Engineering

02-Resistance of ships

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RESISTANCE OF SHIPS

When a ship floating in water is towed by an external means, forces acting in the opposite direction

to the direction of motion will be created both on the underwater part (exposed to seawater) and the

upper part (exposed to air) of the hull. Those forces are proportional to the mass density of the

surrounding fluid, water or air. However, since the mass density of the water is much larger than that

of the air (about 800-850 times in ambient conditions), only the underwater part is taken into account

(except for cases like fast planning craft and hovercraft). A number of hull form designs have been

made to ensure proper operating conditions, weight, area and space requirements to address various

demands. Following is a summary of the various hull designs, based on the way by which the hull is

supported when operating at the design conditions. Although aerodynamically and aero-statically

supported boats (balloons and aircraft) are not in the area of interest of a ship designer, they are

included in the classification for the completeness of the family tree of the marine craft.





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Single hull, displacement type vessels represent about 99 % of the commercial and more than 70 %

of the existing military vessels. Therefore, unless otherwise stated, ships studied in this course refer

to vessels of single hull displacement type.

Components of Ship Resistance

The resistance (or, towing force) of a ship is the force required to move the ship on a straight course

at a given constant speed. This total resistance can be analysed by dividing it into several

components. These components are influenced by a number of conditions, and interact with each

other in a complex way.

For engineering purposes, the calm water resistance of a hull on a straight course and in an

unconfined mass of water can be divided into two main components:

1. Bare hull resistance

2. Appendage resistance

Bare hull resistance is the main component of the resistance in most ships. It can be assumed that it

is the sum of four main components:

a. Frictional resistance:, A ship experiences a frictional force between its exterior underwater hull

surface (“wetted” surface) and the surrounding water as it advances on a straight path. This

component stems from the friction between water particles and the surface of the hull.

b. Wave making resistance: As a ship advances in water, it creates a moving pressure field around

it. This pressure field, which moves together with the ship, causes the free surface of the sea to

be disturbed. These disturbances of free water surface propagate away from the ship in forms of

a wave system. An energy is required for the creation of this wave system. The energy expended

for the creation of those waves can be accounted as the source of the so-called “wavemaking

resistance”.

c. Eddy resistance: An ideal inviscid fluid flows around a submerged body in smooth streamlines.

However, an actual fluid (water for the case of underwater hull and air for the case of

superstructure and fittings) cannot always flow along those smooth streamlines: A boundary

layer, due to friction with the water particles and the hull surface is formed. This boundary layer

is turbulent in nature over most of the length of the hull. In areas around the hull surface where

the water particles are forced to experience a rapid slowing down (i.e., decelarate), these particles

in the boundary layer cannot follow the flow contour and separate, forming a number of eddies.

An observer at a location close to the stern of a ship or a small boat can easily see those eddies.

The formation of those eddies require a sizeable amount of energy. Eddy resistance arises from

the energy expended in forms of eddies about the hull.





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d. Wind (air) resistance: As air flows about the above-water part of the ship, i.e., about the main

hull, superstructure and the deck fittings, frictional and eddy resistances form with mechanisms

similar to that of the water-related resistances. Although wind resistances of classical hullforms

are much smaller than the water resistance, wind resistance can reach to higher values for the case

of fast vessels: It can be a major component of the resistance in hydrofoil, hovercraft or wing-in-

ground effect type vessels.

e. Due to their nature, frictional and eddy resistances are summed up together and called “viscous

resistance”. Therefore, total resistance is the sum of viscous and wavemaking resistances. For

fast vessels, air resistance should also be included.

f. Appendage resistance is the resistance of components other than the hull itself: rudder, shaft

brackets, shaft, propeller, nozzles and flow regulating devices, bow and stern thrusters, sonar

dome, bilge keels, roll stabilizers, etc. These resistances are mainly viscous in nature. In

practice, appendage resistance of each component is calculated separately and added on the bare

hull resistance. Appendage resistance gains importance especially on warship hull forms.

Dimensional Analysis

Dimensional analysis is the most invaluable tool of the engineer searching for relationships between

the interactions of various factors influencing a complex situation. A typical example is the

resistance of a ship in calm water.

The resistance force a ship proceeding in the sea at calm weather encounters is, related intuitively

to(R, dimension: MLT-2):

The density of seawater, (dimension: ML-3)

Speed of the ship in relation to the mass of seawater, Vs (dimension: LT-1)

Characteristic dimensions of the ship, where the waterline length of the ship is a good

representative, Ls (dimension: L)

Gravitational acceleration: The waves that are propagated on the surface of the sea are mainly

produced by the difference in the specific weights of the air and the water. In practice, the

specific weight of the seawater is about 800 - 850 times that of the ambient air and only the

specific weight of the seawater is taken into account. = g, and since was taken into account

formerly, g is taken. (Dimension: LT-2)

Viscosity of the seawater, . However, = , and also since that was taken into account

formerly, the kinematic viscosity is taken into account. (Dimension: L2T-1)

a b c d e

s sR V L g





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MLT-2 = (ML-3)a (LT-1)b Lc (LT-2)d (L2T-1)e

Solving for the powers,

MLT-2 = Ma L-3a+b+c+d+2e T-b-2d-e

or,

a = 1

-3a + b + c + d + 2e = 1

-b - 2d - e = -2

This system does not have a unique solution since it has three equations and five unknowns.

However, any solution should comply the relations:

a = 1

b = 2 – 2d - e

c = 2 + d - e

Hence, the relation for the ship resistance becomes:

2 2 2 1d e

s sR V L gLV VL

where the powers d and e are to be determined by experimental means. Ls2 is directily proportional

to the “wetted” surface area of the hull for “geosim” (geometrically similar) hulls. Therefore, one

can write:

2 212

,s s s

s s

gL V LRf

V S V

The right hand side of this last equation can be lumped as a dimensionless coefficient called

“resistance coefficient”, or, “total resistance coefficient”, CT and:

212 s TR V S C

Noting that the two main parameters of CT are the square of the inverse of the Froude number and the

Reynolds number,





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s

s

VFr

gL

Re s sV L

and hence:

Re,FrfCT

it can be concluded that the two geometrically similar models (geosims) of different sizes, but having

the same Reynolds and Froude numbers do have the same total resistance coefficients, CT.

Unfortunately, this is not possible unless the two geosims are identical. In other words, to test the

entire hydrodynamic properties of a 350 m. tanker we should have a model of 350 m, which is not

possible. This difficulty can be avoided by a testing procedure applied in present day towing tanks,

developed after the pioneering work of Sir William Froude in the second half of the 19th century.

Despite the latest developments in numerical procedures, hull forms are developed by a number of

hydrodynamic testing procedures involving towing tests in model tanks.

Model testing in towing tanks

For a ship and its geosim model to be tested hydrodynamically in a test facility to have the same

component of resistance, either Reynolds or Froude numbers should be equal. Denoting the ship and

the model with the subscripts s and m, for the equity of the Reynolds numbers, i.e., for the equity of

inertial forces to viscous resistance components (sum of frictional resistance and eddy resistance

coefficients)

Re Res m

s s m m

s m

V L V L

or,

m s m

s m s

V L

V L

If the testing environment is water, m = s and:

mmss

m

s

s

m LVLVL

L

V

V or





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For the equity of the Froude numbers, i.e, for the equity of wavemaking resistance coefficient,

s mFr Fr

s m

s s m m

V V

g L g L

or since gm = gs,

m

s

m

s

L

L

V

V

If the Reynolds numbers of the ship and its model have to be the same (Reynolds number similarity)

assuming that the test is made in water as well, ensuring s = m and a model scale of 10s mL L

is to be used, one gets the result Vm=10 Vs. In other words, the model should be towed in a tank at

100 m/s if the condition of Reynolds number similarity for a ship speed of 10 m/s is to be met.

On the other hand, if the Froude numbers of the ship and the model are to be the same (Froude

number similarity), assuming the acceleration of gravity is the same both in the sea and at the

location of the testing facility, ie, gs = gm, one gets the testing speed for the model as Vm = Vs / 10 .

In other words, the model should be towed at 3.62 m/s if the condition of Froude number similarity

for the ship speed of 10 m/s should be met.

Among the various parameters involved, a parameter that is most sensitive to the environmental

conditions is the kinematic viscosity of the water. In terms of temperature (t) (in centigrades) and

salinity (s) (in percentage), kinematic viscosity (in m2/s) can be expressed as:

610 0.014 0.000645 0.0503 1.75s t t

For daily calculations of ship hydrodynamics, a well-accepted value for kinematic viscosity of

seawater is = 1.188 10-6 m2/s and = 1.139 10-6 m2/s for fresh water, both being values at 15

oC. A well-accepted value for the acceleration of gravity at sea level is g = 9.80665 m/s2.

Froude’s testing procedure

Sir William Froude, back in 1868, has developed a testing procedure, assuming that the coefficient of

resistance of the ship is composed of two components:





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Coefficient of frictional resistance, CF : This coefficient is the coefficient of resistance of a

straight, smooth flat plate having the same length as the ship and towed in the sea at the same

speed as the ship. It is obvious that CF is a function of the Reynolds number only.

Coefficient of residual resistance, CR : It is assumed that this coefficient is a function of the

Froude number only. It is obtained by the Froude number testing of a small scale model of the

ship and subtracting the coefficient of frictional resistance of the smooth flat plate having the

same length as the model. It is assumed that the model and the ship have the same coefficients of

residual resistance. It is obvious that the coefficient of residual resistance shall include

wavemaking and eddy resistance components, and shall also include the differences between the

frictional resistance of a flat plate and the actual hull.

EXAMPLE PROBLEM:

A 100 m. long monohull, displacement type frigate is to be tested by Froude’s testing procedure. A

1/20 scale geosim model is prepared and towed at various speeds representing the speeds of the ship

from 3 to 30 knots. Underwater surface area of the bare-hull ship is 1800 m2. Water in the towing

tank is freshwater at 20oC and the seawater salinity is 2.5%, temperature is 18oC.

(a) Find the towing speed of the model corresponding to the ship speed of 25 knots.

(b) If the force measured to tow the model at this speed is 110 Newtons, estimate the resistance of

the prototype ship by Froude’s method

(c) Estimate the ideal power to propel the ship at that speed.

Solution:

(a) The ship model should have the same wave-making characteristics as the “prototype” ship, ie.

Frs = Frm

s m s m m sm s

ss m s m

V V V V L VV V

LgL gL L L

255.59 2.876 /

20m

kntsV knts m s

Where: 1 knt = 1 nautical mile (nm)/hr (1 nm = 1852 m., 1 hr = 3600 s)

(b) Underwater surface area of the model shall be 1800/2 = 1800/202 = 4.5 m2. Therefore, the total

resistance coefficient of the hull shall be:





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2 21 3 1 2

2

1100.005911

0.5 1000 / 2.876 4.5

TmTm

m m

R NC

V S kg m ms m

The viscous component of the resistance can be found from the Reynolds number of the model:

Re m mm

m

V L

For the freshwater model tank at 20 oC, m = 1.00210-6 m2/s. The Reynolds number of the model

shall hence be Rem =1.4351 107. The coefficient of friction of the hull surface shall then be

estimated from the so-called ITTC line (an empirical relation, to be discussed later):

2

10

0.0750.002820

log Re 2Fm

m

C

The coefficient of residual resistance of the hull shall hence be:

0.005911 0.002820 0.003091Rm Tm FmC C C

It is now hypothesised that the residual resistance coefficients of the model and the prototype ship are

the same, ie.,

CRs =0.003091

The kinematic viscosity of the seawater (s =2.5) at 18 oC shall, from the relation above be

s=1.03710-6 m2/s. The Reynolds number of the prototype ship shall be:

9

6

(25 1852 /3600) 100Re 1.240 10

1.037 10

s ss

s

V L

Therefore, using the same ITTC relation for the coefficient of friction of the protoptype ship’s hull,

2

10

0.0750.001491

log Re 2Fs

s

C

And the total coefficient of resistance of the ship shall then be:

0.001491 0.003091 0.004582Ts Fs RsC C C

The total resistance of the ship shall be:





9

212

23 212

1025 25 1852 ./ / 3600 / 1800 0.004582 699.163

s s s s TsR V S C

kg m knts m knts s hrs m kN

(c) The “ideal” power to propel the bare-hull ship at that speed shall be, with no energy loss to the

sea or air by the propulsion system is termed as “effective power” (as shall be outlined in the

following chapter) and shall be:

1

, 699.163 12.861 8992 9000s i e Ts sP P R V kN ms kW kW

Shortcomings of Froude’s testing procedure:

1. The hull of the ship is assumed to be a rectangular flat plate, having the same length as the ship’s

waterline length. This is not true since the hull is a three-dimensional body with variations both

on the shape of the wetted underwater surface, causing divergence or convergence of streamlines

and has significant pressure gradients along the streamline.

2. The hull of the ship and the model are not smooth in hydrodynamical sense.

3. The wave-making components and the eddy-resistance component of the hull are lumped into the

so-called “residual resistance”. Wave-making resistance is mainly a function of the Froude

number and the ship’s geometry. Eddy resistance is a function of the ship’s Reynolds number

and the ship’s geometry.- like adding apples and pears together.

As can be seen, Froude’s testing procedure is a crude approximation and is discontinued in modern

towing tanks. However, it was a historical breakthrough in ship science.

ITTC 1978 Method:

International Towing Tank Conference (ITTC), has developed a procedure taking the criticisms of

the Froude method listed above into account. They have re-divided the residual resistance coefficient

of Froude into wave-making and form (or eddy resistance) components:

Re,Fr Fr ReR W formC C C

Although lacking physical exactness (wave-making resistance is influenced significantly by the

boundary layer formation on the hull, which is dependent on Reynolds number and hull form and

frictional resistance is influenced by the waveform on the hull to some degree), this procedure is

accurate enough for engineering purposes. Therefore, the total resistance coefficient of the hull shall

be:





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Re Re FrT F form WC C C C

The frictional and form resistance coefficients, both being functions of the Reynolds number, are

often lumped together in form of “viscous resistance coefficient” and the form resistance is assumed

to be a constant multiple of the coefficient of frictional resistance:

1v F form FC C C C k

Where:

The factor “1+k” : is termed as “form factor” correction.

Form factor k can be found from the analysis of the towing tank results, as well as from various

empirical formulae for the form factor. The results in the literature reveal that the form factors are

the functions of coefficients of form (CB, CP, CM, CA, CVP) of the hull. Typical values can be as high

as 1.60 (for small, full form vessels such as slow speed barges), where for tankers (CB 0.80) k =0.40

is a typical value. It can be as low as k = 0.05 for slender, fast ships like destroyers (CB 0.45).

Small marine craft with high beam to length (B/L) ratios also have high coefficients of form. A first

approximation for form factor can be found from the empirical formula by Watanabe:

2

25 60 095 B. C

k .L B

B T

An experimental procedure using towing tank results is developed by Prohaska, which shall be

outlined later in this chapter.

Based on the discussions above, International Towing Tank Conference has developed an empirical

procedure that can be outlined as follows:

1. Construct a small scale geosim model for the ship, with a scaling ratio of =Ls/Lm. Test the

model in the towing tank with a set of speeds corresponding to the real ship, with corresponding

speeds yielding the same Froude number (Fr = Vs/(gLs)1/2 = Vm/(gLm)1/2 ). Obtain the coefficients

of resistance CTm for the set of speeds, Vm.

2. From the set of results corresponding to low speeds, deduce the form factor (By the method

proposed by Prohaska, which shall be described later).

3. Find the coefficient of friction of the model hull from the ITTC relation:





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2

0.075

log Re 2Fm

m

C

4. Find the values of model wave-making resistance coefficient:

(1 )Wm Tm FmC C k C

5. It can be deduced that CWm = CWs. Therefore, the coefficient of total resistance of the bare-hull

ship can be obtained from:

6. Resistance coefficient for the appendages can be added to obtain the total coefficient of

resistance for the appended hull. However, in order to account for the inaccuracies incurred

during the testing procedure, a factor called “correlation allowance” can also be used.

Correlation allowance (CA) has a typical value of 0.00040. This correlation allowance takes the

roughness of the hull and the laminar area near the bow of the model, etc. into account.

Therefore,

1Ts Fs W App AC k C C C C

7. The resistance of the full-scale ship and the ideal power required to propel it at a given speed

(effective power) can be calculated.

212

312

Ts s s Ts

e Ts s s s Ts

R V S C

P R V V S C

EXAMPLE PROBLEM:

A 100 m. long monohull, displacement type frigate is to be tested by ITTC 1978 testing procedure.

A 1/20 scale geosim model is prepared and towed at various speeds representing the speeds of the

ship from 3 to 30 knots. Underwater surface area of the bare-hull ship is 1800 m2. Water in the

towing tank is freshwater.

(a) If the force measured to tow the model at a speed corresponding to 25 knots is 110 Newtons and

at the corresponding speed to 3 knots is 1.24 Newtons, estimate the resistance of the ship by

ITTC 1978 method.

(b) Estimate the ideal power to propel the ship at 25 knots speed.

WsFPsFWvTs CCkCCC ,1





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Assume that the total (lumped) drag coefficient of the appendages are 0.004000 and their total

surface area is 50 m2.

Solution:

From the previous problem, it is known that the model is to be towed at a speed of Vm =2.876 m/s

and the coefficient of resistance is CTm =0.005961.

In order to separate the total resistance of the hull, one needs to know the form factor. In the absence

of other experimental results, following logic is used:

“At low Froude numbers, wavemaking resistance vanishes, where the entire model resistance is

almost viscous in nature. Therefore, the total resistance coefficient can be taken equal to viscous

resistance coefficient at low Reynolds numbers.”

For 3 knots, corresponding model speed shall be:

Vm= 3 /201/2 = 0.6708 knots = 0.3451 m/s.

and the model hull’s Reynolds number:

Rem = Vm Lm/m =1.7722 106

Therefore, the coefficient of friction shall be, assuming an all-turbulent friction over the model hull

(!) shall be:

CFm= 0.075/(log10Re-2)2 = 0.004180

On the other hand, total resistance coefficient, from the measured values, shall be:

CTm = RT/(1/2 Vm2Sm) =1.24/(0.5 1000 0.345124.5) =0.004628 Cvm

From which the form factor is obtained to be equal to:

1+ k = Cvm/CFm =0.004628/0.004180 =1.1072

Now, one can revert to the model at speed corresponding to 25 knots, where, from the previous

problem:

Vm = 2.8758 m/s, Rem=1.4351 107, CFm= 0.002820, CTm = 0.005911

Coefficient of model viscous resistance shall be:

Cvm = (1+k) CFm =1.1072 0.002820 = 0.003122





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In the absence of surface roughness of the model surface and appendages, wavemaking resistance

shall be:

Cwm= CTm – Cvm = 0.005911 – 0.003122 =0.002789

Now, one can proceed to the full scale ship. For Vs= 25 knots (12.861 m/s),

Res = 1.240 109, CFS = 0.001491

Cvs = (1 + k) CFS = 1.1072 0.001491 = 0.001651

Coefficient of wave-making resistance, equal to that of the model:

Cws =Cwm = 0.002789

Therefore, the bare-hull resistance of the ship shall be:

CTs = (1+k) CFs + CA+ Cws= 0.001651 + 0.00040 + 0.002789 = 0.004840

RTs= ½ Vs2SsCTs = 0.5 1025 12.8612 1800 0.004840 = 738.518 kN

However, the resistance of appendages (either separately for each or by a lumped approximation)

should be added.

RApp =1/2 Vs2 SApp CApp = 0.5 1025 12.8612 50 0.004000 = 16.954 kN.

The resistance of the ship, by the ITTC 1978 method, is:

RTot= 738.518 + 16.954 =755.472 kN.

(b) The power required (“effective” or “tow-rope” power) is:

PE = RTVs= 9716.21 kW

The determination of effective power is usually the first step in the design of the propulsive system of

the ship.

DETAILED DISCUSSION OF RESISTANCE COEFFICIENTS

Frictional Resistance:

The source of frictional resistance is the viscosity of water. To understand the frictional resistance,

one has first to understand the frictional resistance of an idealized ship, a flat plate (“equivalent flat

plate”), having the same length and surface area, moving in the water with the same velocity U as the

ship. Water particles adjacent to the surface of the flat plate moving in water will “stick” on the plate

surface and will be dragged together with the plate. At some distance vertically away from the plate,





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water will attain a velocity less than the ship’s speed U, increasing with the vertical distance from

the wall and finally will be asymptotically equal to U, i.e., not be influenced by the existence of the

plate. This can be analyzed by assuming a zero-velocity flat plate and a flow of water with a

freestream velocity of U. The layer of water away from the surface of the flat plate(y =0) up to the

location where the u(y) U condition is satisfied is called “boundary layer”. The distance

associated with the condition u() U is satisfied is called “the boundary layer”. The momentum of

water lost within the boundary layer is the origin of frictional resistance. In practice, frictional

resistance of the ship is influenced by a number of effects- Reynolds number, surface roughness,

pressure gradients, three dimensional effects, waves created by the ship....

yU

u(y) =0

U(y) U

Bo

un

dar

y L

ayer

Definition of a boundary layer

By the nature of the flow, boundary layer flows can be classified into two:

1. Laminar boundary layers: Laminar boundary layers are described as the boundary layers where

the flow of water particles take place smoothly, with a given order. This has been explained by some

authors as the flow of “sheets” or “laminae” of fluid at different speeds. As the sheet adjacent to the

wall satisfies the condition u(0) =0 and the layer at the edge of the boundary layer u() = U, the

different laminae are assumed to slide each over the other. As each of the laminae slide on the other,

shear forces occur opposing the motion. This is accounted by the loss of fluid momentum, and is

accounted by a drag force opposing to the flow. This drag force is the “laminar friction” and is equal

to the shear force on the lamina adjacent to the wall. Laminar boundary layers lose the stability

described above under certain conditions. In general, laminar boundary layers are stable for:

Low velocities

Short distances





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High kinematic viscosities

This condition dictates that the quantity Ux/ is small. This is the Reynolds number, alternatively

defined as the non-dimensional ratio of inertial forces to laminar forces within the boundary layer.

The laminar boundary layer loses its stability at high Reynolds numbers and in the presence of

external disturbances. A well-accepted criterion for transition from laminar to non-laminar (i.e.,

turbulent) flow is Re 5105. The stability of laminar boundary layers is also dependent on pressure

gradients on the body.

Obviously, for the equivalent flat plate representing the ship, at the familiar speeds involved, laminar

flow region is small- about 8 cm. from the bow for a 15 knot speed. However, for a small ship model

tested in a towing tank at lesser speeds, laminar boundary layer extends further on the bow relative to

ship’s length and can cause erroneous results. This region is minimized by inducing an early

transition on the model - by sandpaper roughening of the bow, by the usage of studs mounted on the

bow or, by the usage of a trip-wire along the depth before the bow.

Laminar boundary layers are also important in propellers, where the dimensions are smaller than the

vessel by one or two orders of magnitude.

2. Turbulent boundary layers: One can safely assume that the entire boundary layer on the full-

size (prototype) ship and the equivalent flat plate are turbulent, by neglecting the contribution of the

short laminar region. Turbulent boundary layers are characterized by the externally chaotic nature of

the flow, where the order between laminae is totally lost and the flow is characterized by the mean

flow and the flow of eddies imposed on it. The energy of the mean flow is dissipated in form of

eddies, and the associated loss of momentum of the flow is accounted by higher values of friction

when compared with the laminar flow.

The transition from laminar to turbulent flow is not abrupt- small patches of turbulence are created in

the laminar region initially, and in the areas of higher Reynolds numbers, turbulent patches increase

in size and finally dominate the entire flow.

Turbulent boundary layer can be represented by a “power law”:

ny

U

u1





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where u is the mean velocity within the boundary layer, y is the distance from the wall, U is the free-

stream velocity, is the boundary layer thickness. The power n has a value between 7 and 9.5,

increasing with Reynolds number and affected by surface roughness. Coefficient of friction in

turbulent flow is computable by a number of “correlations”, i.e., empirical formulae.

The frictional shear force, w of the flow on the flat plate can be expressed conveniently by a

coefficient of friction, Cf:

21

2w fU C

and the overall drag force on one side on a flat plate by the overall coefficient of friction, CF:

L

f

L

f

FdxC

LLU

dxCU

LU

DC

02

21

0

2

21

2

21

1

The overall coefficient of friction of flat plates have been investigated by several authors in the past-

where reported as a function of the Reynolds number of the plate.





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Coefficient of friction of a flat plate for laminar and turbulent boundary layers

Coefficient of friction of a flat plate, as adopted by the ATTC (Schoenherr) and ITTC (1957 Madrid)

conferences and two authors

Reynolds numbers involved for ships can be high- for a large and fast ship (aircraft carrier or

containership), it can be in the range 3 ~ 5 109. For this range, correlations based are used to

predict the hull frictional resistance. One well-accepted is the Schoenherr formula, accepted by the

American Towing Tank Conference in 1947:

10

0.242log ReL F

F

CC





18

However, a relation later adopted by the International Towing Tank Conference in 1957 that was

more general and more widely accepted is:

2

10

0.075

log Re 2FC

Although these coefficients represent flat plate coefficients of friction, they are for hydrodynamically

smooth surfaces. The additive constant CA= 0.00040 , which is added to the viscous resistance

coefficient CV is a rather crude approximation for the effects of hull surface roughness and for the

inaccuracies involved by the model to ship extrapolation procedure outlined above.

Viscous pressure (Form) resistance:

To understand the viscous pressure resistance, one should know about D’Alambert’s paradox - one of

the main concepts of classical hydrodynamics, expressed literally as:

“For a body immersed in a uniform flow of an inviscid fluid, pressure shall rise in the areas where

the flow slows down and shall increase in the areas where the flow accelerates, according to the law

of Bernoulli. The in-flow direction components of the pressure forces created on the fore part will

have a net positive (backward) drag effect. On the other hand, the high pressure area in the stern

will form negative-drag forces (forces in the forward direction). Those two forces are equal in

magnitude”

D’Alambert’s paradox states that the two net drag forces (acting in forward and backward directions)

will cancel each other and the body shall experience no drag. However, due to viscosity of water,

skin friction and eddy formation behind the body, there is a significant drag component. Equivalent

flat plate approach can be used to calculate this drag force partially.

For all streamlined bodies moving in water, talking with the ship terminology, the bow and stern are

the areas of high pressure and the amidships area is a low pressure area. Therefore, for the forward

quarter of the ship, the flow will encounter a favorable pressure gradient - from high pressure area to

low pressure area. On the contrary, the stern quarter, especially the stern of the ship, shall encounter

an unfavorable pressure gradient - the fluid will be forced to flow from low pressure to high pressure

area. This is associated with the drop of local shear force close to the stern and then afterwards, the

reversal of the flow within the boundary layer. The complicated flow will therefore shall not be

attached to the wall and separate, leaving a flow area with eddies at the stern. This flow area, with

the velocity defect is called “ship’s wake”. For a ship with a separated flow, equivalent flat plate





19

approach is not sufficient to account for the viscous drag force for reasons explained above. The net

difference shall be a force-directed sternwards, called the “viscous pressure drag”.

Boundary layer on a hull

Boundary layer velocity profiles on the hull

An important component of eddy viscosity is the formation of “bilge vortices” at the stern of the

ships, especially full form ships or ships having sterns with rapid changes of curvature. The energy

of bilge vortices are accounted by extra values of viscous pressure drag, as well as they are the causes

of uneven wakes- a cause of propeller-induced vibrations and propeller inefficiency.





20

Formation of bilge vortex

Also, the equivalent flat plate approach the representation of the ship’s underwater surfaces by the

equivalent flat plate- involves a number of inaccuracies: deviations from the flat plate shape, causing

divergence or convergence of streamlines, pressure gradients, crossflow effects on the hull, change of

the underwater shape by the surface wave profile on the ship... cause extra additions to the coefficient

of friction.

Viscous pressure gradient, being a function of the Reynolds number and the boundary layer

formation on the hull, is hypothesized to be a constant multiple of the frictional drag of the hull, is

expressed as:

VP vp FC k C

Lumping the three dimensional effects on the hull together, the viscous resistance coefficient

becomes:

1V FC k C

where (1+k) is the “form coefficient”. Form coefficient is assumed to be unaffected by the Reynolds

number (scale and ship speed) and hull roughness effects. In practice, obtaining of the form

coefficient is made by extrapolating the total ship resistance measurements to zero speed (where the

wave making resistance becomes equal to zero).

F

T

FnF

Fn C

C

R

Rk limlim

00

1

A well-accepted procedure is the Prohaska’s method, where the total resistance coefficient of the

ship model normalized by the coefficient of resistance of the equivalent flat plate (CT/CFo) is plotted





21

as the ordinate and the abciassa is the fourth power of the Froude number, also normalized by the

coefficient of resistance of the equivalent flat plate (Fn4/CFo).

Prohaska’s method for obtaining the form factor

Wave making Resistance

Pressure variations on the hull surface are also the reason for the formation of surface waves created

by the ship. The pressure rises about the bow and stern parts of the hull compared with the still water

pressure. These high pressure areas cause rise of the free surface in those parts, which propagate

away from the hull in the form of a wave array. The energy carried away from the ship in form of

waves is the origin of wave resistance. The outward energy flux of the waves at a control volume

surrounding the ship at distances sufficiently away from the ship, divided by ship speed is defined as

the wave making resistance.

waveW

s

ER

V

A simple explanation of the wave creation is by the assumption of a simple, streamlined, ship-like

body moving in an ideal (inviscid) fluid, applicable both for gas and liquid flows. The velocity of

fluid moving on the body shall be “zero” at two “stagnation points”: one on the forward end and one

on the rear end. The regions near the stagnation points shall have velocities of flow less than the

velocity of body. According to the law of Bernoulli, those low velocity areas shall have pressures

above the ambient pressure. The velocity of flow near amidships shall be higher than the velocity of

the body, and hence the pressure there shall be lower than the ambient. The pressure areas forward

and astern will sum up to force values that are equal and opposite to each other and hence the

resistance in an infinite, inviscid fluid shall be equal to “zero” (D’Alambert’s paradox). However, if

the body is moving on the free surface of a liquid, all points along the streamline shall have the same





22

pressure, the ambient atmospheric pressure. In this case, high pressure areas (bow and stern) shall be

associated with a rise in free surface, and low pressure areas (amidships) by a drop in free surface,

also in accordance to the law of Bernoulli:

2

constant2

v

g

Therefore, there shall be a wave system travelling together with the ship-like body: A wave crest in

the bow, a wave trough in the amidships and another wave crest at the stern. This is called “the

primary wave system”.

Pressure and velocity fields giving rise to the primary wave system of a streamlined body

In real ships, the high and low free surface distributions created by bow and stern are radiated

outwards from the ship, and the wave pattern will be highly dependent on the Froude number.

The first scientific investigations are owed to Lord Kelvin and Dr. Michell, in the late 19th century.

Lord Kelvin has formulated the wave pattern of a simple point source travelling on the free water

surface. This wave pattern is the well-known “Kelvin

wave pattern”, as can be seen in the figure below:





23

Kelvin wave pattern

As can be observed from the figure above, two series of waves are created: One set of diverging

waves moving away from the point source and another set of transverse waves following the point

source. Both of these waves are contained within two straight lines, = arcsin(1/3) = 19.5o from the

direction of travel.

Lord Kelvin has assumed that a ship can be represented by a “source”, or by an hypothetical point

that produces an outward flow of water (i.e., a creator of radially outward streamlines).

Mathematical analysis shows that wave making resistance of this “Kelvin source” shall be:

2 22 3

2

1cos

2WR V A d

Where:

A() : is the wave amplitude.

One can deduce that RW A2Vs2, and also AVs, RWVs

4. The behavior of a ship in the far field

obeys the Kelvin pattern, although it is inaccurate to model the ship by a point source in the near

field.

As this simplified analysis indicates, frictional resistance increases with roughly the square of the

ship speed, while wave making resistance increases with the fourth power of the ship speed (rather,

the Froude number). This fact is more complicated by the fact that the bow, stern and

discontinuities on and slightly below the waterline surface have wave patterns of their own,

interacting with each other. Those individual waves are sometimes augmenting, sometimes

cancelling each other, depending on the shape of the hull and the Froude number. Roughly, a ship

wave system is the superposition of:

The bow- beginning with a wave crest

Forward shoulder – beginning with a wave trough

After shoulder – beginning with a wave trough

The stern- beginning with a wave crest





24

Bow and stern waves on a ship, wave making and frictional resistances (V in knots, L in feet)

The most prominent of those waves is the bow wave. Bow wave is mainly determined by the half-

angle of entrance of the free water surface. Therefore, whenever high wave making resistance values

are a challenge for the ship designer, the main option should be lovering the entrance angle of the

vessel - as is the case for various warship and fast commercial (passenger ship, containership, etc.)

forms. On the other hand, sleek entrances lower the deadweight capacity of the ship. Therefore, in

ships that have to carry cargoes at acceptably lower speeds, sharp bow shapes are replaced by almost

cylindrical or bulbous bows. Bulbous bows have the ability to reduce wave making resistances at

relatively lower Froude number ranges. (Less than Fn = 0.25). Many of present day cargo ship forms

are characterized by either cylindrical or bulbous bows or higher coefficients of form (crude oil

carriers, bulk cargo carriers, OBO’s, etc.)

Due to the interaction of those different waves, the total wave system will be highly nonlinear: with

pronounced humps at about Fn =0.30 and Fn = 0.50, and an intermediate trough at about Fn = 0.35.





25

Wave resistance versus Froude number

An analytic expression, more sophisticated than the assumption of Kelvin source for a given hull was

given in 1898 by Michell, for a hull whose fore-aft direction lies along the x- axis and z- is the

vertical ordinate:

22 2

21

4

1wR V A d

2z i x

S

A i e f x,z dz dx

Where:

: is the dummy variable of integration,

V : is the ship speed,

: is the density of seawater,

=V2/g : g is the gravitational acceleration,

S : is the underwater hull surface below the calm water line and

y = f(x,z) : is the hull surface offset from the centerline plane in the y- (athwart ship) direction.

Modern digital techniques model the ship as a number of sources and sinks distributed along the hull

surface on discrete panels. Accurate predictions of wave pattern is possible, but however, mainly due

to the neglecting of viscosity effects, some errors exist. Although partially corrected by empirical

viscosity corrections (boundary layer formation), errors still prevail. Therefore, model towing tank

experiments are still an indispensable part of hydrodynamic analysis of the ships.

Ship wave system also has an effect on the viscous resistance components of the ship, by inducing

velocities along the streamlines and by a non-straight path on the free water surface.

Bulbous bow forms:

Many merchant ships produced during the last few decades have a bulbous bow form. Bulbous bow

form is a means of reducing the wavemaking resistance of the ship, where at the designed service

speed of the vessel, wave trough of the bulbous bow cancels the bow wave crest, thereby reducing

the wavemaking resistance significantly. This effect has been known for centuries, as is visible by

the resistance decrease of ram-bow type warships of the past. A number of bulbous bow shapes have

been produced: Pear-shaped, inverted pear-shaped, elliptical cross section, cylindrical, etc. It is





26

obvious that the desired effect of the bulbous bow takes place at the proper operating speed and at the

proper immersion of the bulb: which means the effect of bulb is also dependent on the loading

condition of the vessel.

Simplified explanation of the effect of bulbous bow on wavemaking

Since the stern part of a ship is also a high pressure area, which is responsible for the formation of

stern waves, a plausible approach shall be the fitting of stern bulbs as well. Although designs

involving stern bulbs have been made, they have not been so popular since the gains obtained are

marginal compared to the bow bulb and the high cost of manufacturing of a stern bulb.

It has also been proposed to replace bulbous bows by wing-like foils that emulate the same pressure

distribution and waveform.

Transom stern forms

Transom sterns are widely used in almost all ship types, replacing the cruiser stern forms. The main

reason for the use of transom stern in merchant ship forms where Fr <0.3 is the ease of manufacture

and hence lower cost. The transoms of those ships are above the waterline, and do not have any

hydrodynamic effect. However, for high speed marine craft, transom stern is for the reduction of

wavemaking resistance. The effect of transom is by increasing the virtual length of the ship and

hence increasing the Froude number. The depth of transom below the waterline should be about 10-

15% for Fr 0.5 and 15-20% for Fr0.5.

Stern wedges and stern flaps

As a ship reaches to speeds corresponding to Froude numbers of about 0.40 and above, bow wave

gains dominance and as a result, the ship trims backward. This results in another component in

wavemaking resistance: the component of ship’s weight in the direction of propulsor thrust. This can

be remedied by fitting of trim-reducing mechanisms. The first one is the stern wedge, fitted to

warship hull forms. The function of the stern wedge is deflecting the streamlines downward to





27

produce an upward force component, hence creating a forward trimming moment to reduce the trim

by stern. The same effect is achieved in high speed planing hulls by hydraulically-controlled stern

flaps.

Stern wedge of a frigate

Apart from the classical viscous, wavemaking, appendage and air resistances, certain components of

resistance on certain hull forms arise. Those are:

Wave-breaking resistance:

Another component of wave resistance is the “wave breaking resistance”, which can be significant in

full form ships (tankers, bulk carriers, barges, etc.) and the bow has a bluff shape. The bow of the

ship can collide with the bow wave before it propagates into the sea, causing it to break and release

its energy. The energy of the broken waves is dissipated by viscous effects. Therefore, wave

breaking resistance is viscous in nature, although it is mainly a function of Froude number.

Wave-breaking phenomenon can be prevented by making the radius of curvature r of the bow form

such as r Vs2/50, where r is in meters and V in m/s.

Wave breaking resistance can be observed in the wakes of such ships, as two visible white rows

extending backwards from the stern (white colour comes from the air bubbles entrained as the wave

breaks).

Spray resistance:





28

In very fast vessels, high pressure at the forward stagnation point causes an upward spray of water,

also resulting in an enery loss that can be significant. This spray resistance, if not prevented by the

shaping of the bow, can be deflected downwards by spray-rails, hence reducing the unwanted effect.

Wind (air) resistance:

The aerodynamic resistance of the upper parts of the main hull and the superstructure, masts, funnels,

antennae, etc., also viscous in nature, can become appreciable at higher speeds and can have a

significant effect both in propulsion and steering characteristics at higher wind speeds. Due to the

boundary layer effect of the sea surface, the higher the component is located, the higher the velocity

becomes, according to the power law:

7.5

ref ref

u z z

U z

where the reference wind velocity Uref is evaluated at the standard reference height zref above the sea

level, usually 10 m.

Wind resistance is a function of the wind direction and ship speed. At calm weather, the wind blows

from the bow and the wind resistance is a function of the ship speed alone. In this case,

21

2wind air s Tr xR V A C

where ATr is the transversal projected area of the ship. A typical value of the wind-resistance

coefficient is Cx = 0.8.

Appendage Resistance:

The principal appendages to the hull are the rudder(s), shaft bossings or shaft brackets, the shaft and

the propeller, struts, bilge keels, roll stabilizing fins, sonar dome, bow thruster.... All of those items

contribute to the underwater resistance and their drags are calculated separately. The calculation

procedure is similar to the hull itself. Each appendage can be modelled by a frictional and a form

coefficient. Therefore, both the wetted surface area and the geometry of the appendage are

important.

The breakdown of components of a bare-hull ship is presented below graphically.





29

Following table shows the relative contributions of resistance components of bare hull resistance:

Type of

resistance

Percentage of total barehull resistance

High speed

ship

(Container

vessel)

Low speed ship

(VLCC)

Friction RF 45 90

Wave RW 40 5

Eddy RE 5 3

Wind Rw 10 2

IN-SERVICE RESISTANCE COMPONENTS

The discussion of the ship resistance explained above applies only to the calm-water resistance of a

ship cruising on an infinitely deep water and on a straight course, without reverting to use her rudder.

Furthermore, the ship is assumed to have a clean and uncorroded hull, as it is newly launched to the

sea and “even-keeled” (without any fore or astern trim). In real life, it is not always possible to

obtain those situations. In-service situations are discussed below.

Wave Added Resistance:





30

A ship in a seaway generally encounters a speed loss. One part of this speed loss is due to captain’s

initiative of lowering the ship speed so as to encounter waves at a lower velocity or change of course

for a more favourable wave direction (“Voluntary” speed loss). The other part is due to the increase

of ship resistance due to the diffraction and reflection of waves, as well as the motions of the ship to

cause the ship deviate from the hydrodynamically favourable position. This is a complicated

problem of engineering, since too many parameters are involved: Wave height, ship speed, relative

direction of waves, ship length relative to wavelength, bow shape, ship hull shape, draft, wind speed

and direction. Approximate formulae for the wave-added resistance are given by Townsin and Kwon

in form of speed loss for a given power as:

shipV C C V

where C is a factor considering the predominant direction of wind and waves, angle measured

from bow, as well as the seastate in Beaufort scale, BN:

C = 1 for = 0-30o

C = 1.7 – 0.03(BN – 4)2 for = 30o – 60o

C = 0.9 – 0.06(BN – 6)2 for = 60o – 150o

C = 0.4 – 0.03(BN – 8)2 for = 150o – 180o

Cship is a factor considering the ship type:

Cship= 0.5 BN + BN6.5/(2.7 2/3) for tankers, laden

Cship= 0.7 BN + BN6.5/(2.7 2/3) for tankers, ballast

Cship= 0.7 BN + BN6.5/ (2.2 2/3) for containerships

Where:

: is the displacement volume in cubic meters.

Shallow Water Added Resistance:

A ship cruising in shallow waters will encounter an increase of resistance. This is due to two effects:

the first one is due to the increase of draft and trimming of the ship (usually by stern) by the effect of

sea floor (“Squat”), and the second one is due to the change of the wave pattern formed by the ship.

Squat can have other dangerous effects- grounding of the ship.





31

The effects of shallow water have been calculated and given in forms of curves. A well-accepted

calculation procedure is due to Schlichting, as given in the following chart, where:

Ax : Maximum ship sectional area (usually equal to CM)

H : Water depth

V : Ship speed in infinitely deep water

Example: A tanker, 300 m. in length, 50 m. in breadth, 25 m. in draft has a midship section area

coefficient of CM = 0.99. Her service speed is 15 knots. She enters to a shallow area of depth 40

meters. Estimate her final speed.

Ax = CM B T = 0.99 50 25 = 1237.5 m2

xA

h0.88

V = 15 knots =7.717 m/s. , V2/gh =0.152

Therefore, speed loss due to shallow water shall be 12 % (approx.), and the speed shall drop to 13.2

knots.

Schlichting’s curves for relative speed loss in shallow water

Steering resistance:

In a real sea environment, with the presence of external disturbances like waves and winds, the ship

is forced out of her straight course. During this turn, the ship experiences a yaw motion, which

increases the frontal area of underwater surfaces to forward motion, and the rudder is used to set the





32

ship back into position - thereby adding a lift-induced drag to the forward motion of the ship. The

result is a sinusoidal motion, and can also be studied by statistical means. Obviously, steering

resistance increases in harsh weather conditions, where the ship is deflected from her original course

by external wind and wave effects.

Hull roughness

Hull roughness can either be in the form of bio-roughness, that is, biological fouling (by underwater

algae and animal forms), or by the deterioration of hull material by corrosion and the hull paint.

A relation to predict the hull roughness effect of a painted hull on frictional and hence on viscous

resistance for large ships is the equation proposed by Townsin for 10-30 knots range:

1 1

3 3 310 44 10Re 0.125FhC

L

where h is the mean apparent hull roughness of 50 mm. long samples. A new ship has roughness

values of about 100-120 microns, while a 25 year old vessel can have about 1000 microns, even if no

biological fouling exists. It can be observed that roughness penalty is proportional to the one-third

power of the relative hull roughness, h/L and gains a relative importance in larger vessels such as

VLCC’s and large bulk carriers.

Example: A LWL = 100 m. ship with a service speed of 16 knts has an average hull roughness of 250

microns. Find the final coefficient of friction.

Re =VsLWL/ = 7.937108

CFsmooth = 0.001575 (by ITTC 1957 relation)

CF= 4410-3[(25010-6/100)1/3-10(7.937108)-1/3] + 0.000125 = 0.000247

CF rough = 0.001822

As can be seen, even a smooth-looking hull surface, with 250 microns of roughness, can have a

relative roughness penalty of 16% due to hull roughness. The roughness of larger (low Froude

number) hulls, where most of the ship resistance is viscous, plays a significant role in ship operations.

Keeping the roughness at a minimum can result in significant savings in ship operations. Needless to

say, the growth of macro-organisms on the hull surface (marine algae, mussels, tubeworms,

barnacles, etc.) will have a detrimental effect on the fuel bill of a marine vehicle.





33

Added Resistance due to Subsurface Waves

In certain areas of the seas, like the Scandinavian fiords or the estuaries of rivers, a stratification in

the water beneath the ship can occur: A layer of low density fresh water can be on the surface and

another layer of high density seawater can be at some distance below the surface. A pressure

disturbance travelling on the free surface, such as caused by a slow moving ship, will create a system

of internal waves between the two subsurface layers, taking their energy from the vessel. Those

internal waves shall have an amplitude (in height) that is larger than the surface waves that the vessel

creates, thereby increasing the ship resistance a few times than it should be in open seas. Fortunately,

these waves cannot travel faster than about (say) 5 knots and a vessel travelling more than five knots

will very likely not be affected by those subsurface waves.

Ice-breaking Resistance:

Icebreaker ships, as well as the vessels that are required to operate in the Arctic and Antarctic areas,

or inland waterways of Europe and North America during winter months, have often to navigate in

seas where the surface of the sea is covered with a sheet of ice. The thickness of this ice sheet can

vary and those ships are classified separately with more stringent safety requirements by the

classification societies depending on the approximate thickness of permissible ice thickness. The

bow form of those vessels are designed to break this sheet of ice and open way for the vessel itself or

the vessels to follow it. In such cases, the wave pattern of the ship will be supressed and another

resistance component, “icebreaking resistance” shall arise. The detailed analysis of this ice-breaking

resistance is complex, involving complicated mechanisms of fracture mechanics and often requiring

towing tests performed in towing tanks with frozen free surfaces.

PRELIMINARY ESTIMATION OF SHIP RESISTANCE

Although the most accurate way of determining the resistance of a given hull form is by a series of

model tests, it is usually sufficient to make an estimate of ship resistance by empirical methods in the

preliminary design phase. Model tests are costly and time-consuming. However, there are methods

for the preliminary estimate of ship resistance with acceptable accuracies in the preliminary design

phase. Those methods can be classified as:

Methods that are based on systematic model series

Methods that are based on model experiments

Methods that are based on systematic model tests are based on a series of hull forms, where main

parameters are systematically varied and the variations in resistance parameters are expressed in





34

forms of curves. For a given design project, a suitable series is selected, main parameters related to

ship size, deadweight, speed, etc. are determined and the residual resistance is estimated by

interpolation from a series of curves. Frictional resistance can be calculated from the ITTC 57

formula. Examples of such series are the Taylor, Series 60, SSPA, Taggart, NPL, NTUA and ITU

Fishing Boat series. Since the number of parameters relevant for a ship form are excessive (CB, CP,

CM, CA, CVP, L/B, B/T, LCB, LCF, parameters related to bulb, transom and stern form, half angle of

entrance, length of run, number of screws, number of rudders, etc), only main parameters are selected

(usually CB, L/B, B/T, LCB).

Methods that are based on model experiments are obtained from the databanks containing the

results of model tests of various hullforms. They are also for the calculation of residual resistance

coefficient. Most well-known are the Lap-Keller, Danckwardt, Guldhammer-Harvald diagrams,

Holtrop-Mennen and Hollenbach’s method. The first three methods rely on diagrams, while the last

two relies on empirical correlation formulae. Form factor (k), wave making resistance, and also

ship’s wake coefficient and thrust deduction coefficient is obtained. A simple computer program can

be used to evaluate these formulae for a given hull form.