There are two main components of resistance Viscous resistance

Wave resistance

Wave resistance is associated with the waves generated by a ship alone.

At low speeds (Froude numbers), wave resistance is a small part of total resistance.

At very high speeds (Fn more than about 0.45),wave resistance increases so much thatconventional displacement type vessels cannotgo at such high speeds.

It is necessary to use unconventional hull forms for ships of very high speed.

A three-dimensional body moving in water hasa pressure distribution around it.

In case of a ship, the free surface will rise atregions of high pressure and fall at regions oflow pressure.

A moving pressure distribution will give rise tofree surface waves accompanying the body.

This was observed by W. Froude, who made asketch showing the waves generated by a ship.

The waves generated by a "moving pressure point” were studied by Lord Kelvin. (Figure).

The main features of the Kelvin wave pattern are : The pressure point generates a series of transverse waves and a

series of diverging waves.

The transverse waves are slightly curved convex forward,travel at the same speed V as the pressure point, have a wavelength appropriate to the wave speed (i.e. )

The diverging waves emanate from the pressure point and jointhe transverse waves that lie on lines radiating from thepressure point on either side of the direction of motion atangles of 19o 28’ (sin-11/3).

22 V g

The waves generated by a ship in calm water have the similar features.

The moving pressure distribution of the ship can be regarded as an assembly of pressure points all producing Kelvin wave patterns that are superposed on each other.

There is a bow wave system and a stern wave system, each with diverging and transverse waves (Figure) : Each system lies within lines making an angle with direction of

motion; the angle depends on the hull form. The bow transverse waves start with a crest aft of the bow. The stern transverse waves start with a trough forward of the

stern. The stern wave system is superimposed on the bow wave

system. Diverging waves are steeper and therefore more visible. The combined wave system spreads far and wide behind the

ship with the wave heights reducing and finally dying out.

The superposition of the stern waves on the bow waves results in wave interference that may be favourable or unfavourable.

If the transverse waves from the bow are in phase with the transverse waves from the stern, i.e. the crests of the bow waves coincide with the crests of the stern waves, the resulting waves will have the maximum height, and the wave resistance will be larger than the value it would have if there was a phase difference. Considering resistance, this is unfavourable interference.

If the bow waves and the stern waves are completely out of phase, i.e. the crests of the bow waves coincide with the troughs of the stern waves, the resulting waves have smaller heights and the wave resistance is lower : favourable interference.

Wave resistance does not increase steadily with speed but has undulations. This is shown clearly by a curve of wave resistance coefficient as a function of Froude number, or a curve of total resistance coefficient as a function of speed-length ratio , V in knots, L in feet.

/V L

The “humps and hollows” in a wave resistance curve can also be explained by noting that a deep wave trough at the stern will cause a sharp increase in pressure resistance, while a shallow trough or a crest will reduce the pressure resistance, which is mostly wave resistance. (Figure).

The speeds or Froude numbers at which favourable and unfavourable wave interference (humps and hollows) occur can be calculated.

mL = distance between first crest of bow transverse waves and first crest of stern transverse waves. n = number of complete wave lengths in the distance mL

Values of Froude number, Fn

Hump Hollow Hump Hollow n Speed Speed n Speed Speed1 0.520 0.736 6 0.212 0.2222 0.368 0.425 7 0.197 0.2043 0.300 0.329 8 0.184 0.1904 0.260 0.278 9 0.173 0.1785 0.233 0.245 10 0.164 0.169

(Figure)

The hump corresponding to n = 1 is called the “main hump”, and the hump corresponding to n = 2 is called the “prismatic hump” because it depends upon the prismatic coefficient of the ship.The design speed should preferably lie in a hollow.At low Froude numbers, the wave length is small, there are many waves between the bow and the stern and the wave heights are small. The wave resistance is small and interference effects are negligible.As Froude number increases, the number of waves within the ship length decreases, the wave heights increase and the wave resistance becomes large and the interference effects prominent.As the Froude number approaches about 0.4, the wave length approaches the length of the ship.

Above a Froude number of 0.4, the wave length becomesgreater than the length of the ship, the first trough of thebow waves starts nearing the stern, there is a largedecrease in pressure, particularly towards aft, resulting insinkage and aft trim, and a large increase in waveresistance.

These effects reach their peak at Froude numbers around0.5 (main hump). The sharp increase in wave resistance atFroude numbers above 0.4 acts as a speed barrier for shipsof normal form and unconventional hull forms must beused.

Wave breaking resistance is closely related to wavemaking resistance.In ships with very full forebodies, the flow ahead of the bow becomes unstable and the bow waves break resulting in wave breaking resistance.Wave breaking may be due to flow separation at the free surface ahead of the bow.Another view is that wave breaking occurs when the streamlines at the bow have excessive curvature causing flow instability.Guidelines for avoiding wave breaking include making the radius of curvature of the streamlines sufficiently large (R>V2/50, metric units), and limiting the half angle of entrance and the slope of the tangent to the sectional area curve at the forward end within limits.Wave breaking resistance is not important for most ships.

Appendages :

Single screw ships : rudder, bilge keels and stabiliserfins, skeg

Twin screw ships : shaft brackets or bossings, open shafts, multiple rudders

“Negative appendages” : projections into the hull, e.g. tunnels of lateral thrust units

Appendage resistance from model tests :

Appendage resistance = total resistance with appendages – bare hull total resistance

Difficulties:

Accuracy of small appendages such as bilge keels

Scaling problems : Reynolds number effects, laminar flow

Appendage scale factor

Propulsion devices and their components not considered as appendages

Rudders in propeller slipstream

Empirical formulas for resistance of individual appendages

Preliminary estimates of appendage resistance as a percentage of total bare hull resistance

APPENDAGE RESISTANCE AS

PERCENTAGE OF BARE HULL RESISTANCE

Type of ship Values of Froude number

0.21 0.30 0.48

Large, fast, four screws 10-16 10-16 -

Small, fast, twin screws 20-30 17-25 10-15

Small, medium speed, twin screws 12-30 10-23 -

Large, medium speed, twin screws 8-14 8-14 -

All single screw ships 2 - 5 2 - 5 -

Air and wind resistance RAA : D.W. Taylor’s formula :

CD is the drag coefficient

is the density of air

AT is the transverse projected area of the ship above water

VR is the relative head wind speed

Typically CD = 1.2, = 1.225 kg per m3

1 2

2AA D air T RR C A V

air

air

Hughes method (slightly modified) to determine RAA :

Wind force

VR = relative wind velocity = relative wind direction AL = longitudinal above water projected area AT = transverse above water projected area = angle of wind force to ship centre line

1 2 2

2

1 2 2

2

cos cos

sin sin

D air T R

D air L R

F C A V

F C A V

Effect of wind velocity gradient – depends upon relative magnitudes of ship speed and absolute wind speed

Yawing moment if centre of pressure of windforce and centre of lateral resistance have alarge longitudinal separation : resulting driftangle and use of rudder increasehydrodynamic resistance

Air and wind resistance may be reduced by streamlining superstructure and deck houses –but this works only in a head wind.