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In the solution of this equation there is a solution for the sub critical flow Fh<1 and one for the supercritical case Fh>1. Physically this corresponds respectively to the aerodynamic conditions of subsonic and supersonic flow if the Froude number is replaced by the Mach number U/a because this equation is also the equation governing the linearized steady state two dimensional flow in aerodynamics, with compressibility effects represented by the speed of sound which is a.

Imagine now that this analogy is applied to explain what happens in shallow water and it is referred to (a) in Figure 5.7 where a point representing the vehicle is radiating a wave spreading with the speed of sound. After a certain time step dt, the wave front has got the radius a dt. In meantime, the point with velocity U has travelled over a distance U dt. Since U < a, a picture as on (a) in Figure 5.7 is obtained. We are still in the sub critical region where the V shape depends on U/a.

Figure 5.7 Sound waves (Mach)

Figure 5.7 (c) an example of supercritical flow is shown. If we apply the relation given above between and Fnh, we get the following values for and Fnh, which are also given in Figure 5.3:

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nhF (degrees)0.38 19.470.42 19.470.50 19.480.55 19.500.60 19.620.70 20.300.82 23.700.92 39.320.96 59.450.99 781.00 901.005 841.41 451.73 352.00 303.00 19.47

In a towing tank, the depth is approximately 5 to 6 meters. The maximum speed of the models is approximately 10 m/s. A consequence of this is that speed prognoses based on results from tests in a towing tank should be made with carefulness.

From Figure 5.1, Figure 5.2 and Figure 5.4 and the few examples below it is evident that:Fast ships with lengths up to 40 meters operate at a FN number close to 0.90 and at depths between h/L= 0.10 and 0.30, where the influence of shallow water is marked. However, if the model tests shall be relevant for deep water the test should be made at 2.0h Lwhich means 6 8h m . In other words: The model tests are not always 100% representative for the service condition. A fast ship with a length of 150 meters and a speed of 50 knots operates at FN = 0.67. At this speed the critical depth is 67.4 which gives h/L = 0.45. This shows that a larger an fast ship often may operate near the critical speed in full scale. In model scale the depth in a normal tank is about 5.5m, which gives h/L=0.33. If we study Figure 5.1, it is indicated that tests in the towing tank are valied for shallow water. However if it is required that the model tests shall be valid for "deep water", this can not be fulfilled.A frigate has a length of 100 meters and does often operate at h/L=0.23 and FN=0.435. A normal model length would be 6 meters (h/L =0.92). As seen from Figure 5.2 and Figure 5.4, the wave resistance may be doubled, due to the shallow water effect.

In cases where both Fnh and FN >1.25, the transverse waves are not present any more. Figure 2.46 indicates that the wave resistance in this case is due to divergent waves and decreases with FN. The resistance due to the transom stern becomes more important.

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6 DIRECT MEASUREMENT OF RESISTANCE COMPONENTS

the main point in this section is

to show how it is possible to measure the wave resistance, viscous resistance and resistance due to wave breaking separately, using special measurement techniques.

The waves generated by the ship are distributed in the direction of motion (transverse waves) and out to the sides (divergent waves) (see Figure 2.45). At low and moderate speed most of the wave energy is in the waves following the ship (transverse waves). If the speed is high, most of the energy is in the divergent waves.

It is possible to measure velocities and static pressure at different distances from the surface and out to the sides behind the ship. The measurements can be made with a pitot rake placed at a distance equal to 0.5 L behind the model, as shown in Figure 6.3 and below. During tests with a conventional model, measurements are normally made in a depth down to 1 m and in a width of about 3 m.

In such measurements, one control plane is placed in front of the model and one control plane is placed behind the model. The planes follow the model as indicated in Figure 6.1.

Figure 6.1 Definitions for wave resistance measurement

The total resistance can, as already discussed, be expressed by the change in pressure and momentum between two control planes, one far in front and one behind the model:

(( ( )) (( ( ))TA B

R p u u V dy dz p u u V dy dz (6.1)

where A is the control plane far in front of the model while B is the control plane behind the model.

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If the total resistance is known, and the wave resistance is wanted, the viscous resistance is subtracted from the total resistance. The viscous resistance follows from measurements of the total pressure between the control planes. This is done applying Bernoulli's equation, as indicated in Figure 6.2.

Figure 6.2 Measurement of total head

The definition total head has been applied and is defined as:

2 2 2( )2

H p g z u V v w (6.2)

where u, v and w are velocities induced by the hull in the x, y and z directions and V is thevelocity of the ship.

A has a position so far in front of the ship that these velocities are zero or close to zero.

The total head far in front of the ship is then:

H0 = g z +2

V 2

which gives the following total resistance:

2 2 20( ) (( )

2TB B B

R H H dy dz g z dy dz v w u dy dz (6.3)

The total resistance is according to this, expressed by the velocities induced in the control plane behind the hull. The second link is transformed in such a way that the total resistance becomes:

2 2 2 20( ) ( )

2 2TB B B

R H H dy dz g dy v w u dy dz (6.4)

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where is the wave height behind the boat in control plane B at the breadth y

It is assumed that the flow on the outside of the viscous wake is free from rotation, and that H therefore is constant along the streamline (and that Bernoulli's equation can bee applied).

Inside the wake, the flow has rotation. For such cases it is assumed that the width of the wake

is proportional to12x . Maximum wake (maximum back flow velocity) is proportional to

12x .

With these assumptions:

0( )B B

H H dy dz V u dy dz (6.5)

in the wake where:

u friction velocity and a measure of formation of turbulence and

transmission of momentum due to turbulent fluctuations in the wake

= shear stressThe viscous resistance is expressed as:

0( )VB

R H H dy dz (6.6)

and the wave resistance as:

2 2 2 2( )2 2W

B B

R g dy v w u dy dz (6.7)

H and Ho are measured in different ways. It is for example normal to measure H directly as shown in Figure 6.2, where the total head is measured directly applying a Prandtl tube.

The wave pattern resistance can be measured by measuring the wave elevation in a longitudinal cut along the side of the ship, as sketched in Figure 6.10.

It is also possible to determine the viscous resistance by measuring the velocities and static pressure at different distances from the surface and in the breadth behind the model. The measurements are made with pitot rakes at approximately 0.5 L behind the model as shown in Figure 6.3.

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Figure 6.3 Measurement of viscous resistance using a rake of pitot tubes. The vertical rake of tubes must be traversed across the width of the wake.

As shown before, it is possible to express the viscous resistance with the total pressure behind and in front of the model as:

0( )VB

R H H dy dz (6.8)

where H and Ho can be measured in different ways but it is normal to measure H directly without measuring pressure and velocity separately.

The viscous resistance is registered as a change in pressure and momentum between an undisturbed point in front of the model and a point in the control plane behind the model. The plane is placed so far behind the model that the static pressure no longer is influenced by the local flow.

Figure 6.4 shows as an example results from such measurements for a slender model with a sharp forebody and for a full tanker model. It is observed that two types of curves are present. The tanker model has a top in the middle but also one top on each side. These tops on the sides are not due to the viscous resistance. They contain the energy in the breaking wave, as indicated in Figure 6.5.

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Figure 6.4 Results of wake traverse measurements of two different ship types (Baba)

When the wave is breaking, the energy is transformed in such a way that it may be registered as false viscous resistance as shown in Figure 6.5.

Two components of the viscous resistance are therefore identified:

0 1 0 0( ) ( )v v vA B

R R R H H ds H H ds (6.9)

The first integral is valid for the central part, while the second integral is valid for the region, which is influenced by the wave breaking.

Figure 6.5 Division of wake zone due to viscous resistance and due to wave breaking

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The last component may become large but is not proportional to CF. It is only dependent on FN. In resistance calculations, this component must therefore be added to Cw if Cw has been measured by wave cuts. Figure 6.6 and Figure 6.7 show that good correlation is obtained between resistance from towing of the model and resistance obtained by summing up the components measured separately.

Figure 6.6 Resistance components of a slender ship

Figure 6.7 Resistance components of tankers in ballast.

At MARINTEK, several tankers were tested in a research program on energy saving in shipping. Different fore and after bodies were tested and the wave resistance Rw and the components Rvo+Rv1 measured. Two of the hulls are shown in Figure 6.8 and Figure 6.9,together with the wake measurement results and derived resistance. It is possible on both models to recognise the shape of the waterlines or the frames in the after body. It is observed that Rw+Rvo+Rv1 corresponds well with measured total resistance.

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3

3.5

4

4.5

5

5.5

6

0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22

FN [-]

Resi

stan

ce c

oeffi

cien

ts *1

000

CF

CT

CW

CVCV

Figure 6.8 Frame shape, wake measurement results, and calculated resistance for ship model M32 (Marintek)

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3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22

FN [-]

Resi

stan

ce c

oeffi

cien

ts *1

000

CF

CT

CW

CVCV

Figure 6.9 Frame shape, wake measurement results, and calculated resistance for ship model M49 (Marintek)

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6.1.1 Measurement of the Wave Pattern and Estimation of the Wave Resistance.

We have seen that the wave resistance increases with the square of the wave height, and seen that it is possible to measure the wave resistance by measuring the wave pattern of the ship or the model. In the following, it will be shown that this is also the case if the flow is three dim. Figure 6.10 shows arrangements for such measurements in three- dim flow.

Figure 6.10 Longitudinal cut measurement of wave pattern resistance

According to Havelock (1953), a three- dim wave system created by a ship may be expressed as:

22

0 0

2

22

0 0

2

( . ) ( ) sin sec ( cos sin )

( ) cos sec ( cos sin )

x y S K x x y d

C K x x y d (6.10)

where

( , )x y wave height at a given point (x, y)

= angle between the wave direction and the longitudinal axis of the ship. See Figure 2.44.

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( ), ( )S C = amplitude functions (or constants) determined from analysis of a measured wave system and dependent on the shape of the waterlines of the ship.

02

gKV

g = acceleration of gravity

It is also possible to calculate these functions theoretically for a given ship shape.

Assume two planes one in front of ship and one behind the ship perpendicular to the direction of the ship's course. The plane A is far ahead of the ship. The plane B is far in the rear. At a given time, the wave system is crossing the plane B. After a time step t the wave system has moved to the plane A.

The increase in energy in the fluid between the two planes is equal to an increase in wave energy corresponding to the increase in the area of the free-wave pattern. By introducing the distance x V t between the two planes, Havelock by a lengthy calculation obtained an expression for the time average of this energy as:

323 2

2

2

cos( )1 sin

E x V t A d (6.11)

where2 2 2( ) ( ) ( )A S C

is the square of the wave amplitude.

This increase of energy is attributable to the work W1 performed by the ship as it advances against wave-making resistance, RW:

1 W WW t R x R V t

The second source, W2 is the work performed through vertical plane B, by the fluid through the rear plane. W2 is the energy transfer due to the wave motion. The values of W2 and energy E changes with time according to the phase of the wave. The time average is expressed as follows:

522 3 2

2

2

cos ( )( )2 1 sin ( )

W t V t A d (6.12)

From the law of conservation of energy:

21E x W t W t

the wave resistance is expressed in the simple form:

159

2 22 2 2 3

0

( ) ( ) cos2W

WR E V S C dV

(6.13)

It is observed that the wave making resistance is obtained by an integration of the quantity A2( ) weighed by the quantity cos3 . This shows that the wave making resistance increases little when varies. It is also clear that mainly the amplitude of the transverse wave determines the wave making resistance. As the diverging wave, especially at a large -value, has a small wavelength, it is more visible than the transverse wave when ship's waves are observed from above downward vertically. This is because of the fact that the diverging wave has a steeper slope than the transverse wave. This sometimes tends to give an erroneous impression about the relative importance of these two waves with respect to wave-making resistance.

It is now possible to connect the visible ship's wave directly to the wave-making resistance and to compare theoretical computations with experimentally observed ship's waves.

160