## Afc K2 A A05

Thus for dynamic similarity the model propeller must rotate faster than the corresponding ship propeller in the inverse ratio of the square root of the linear dimensions.

The thrust power is the product of thrust and velocity and for the same Froude number:

PT2 p2

Correspondingly for torque it can be shown that:

The rado of torques for geometrically similar propellers at the same advance coefficient and Froude number will be as the fourth power of the linear dimensions. That is:

Coefficients for presenting data It has been shown that:

Substituting y, = NDJ in these expressions:

J2 [/(/) ] is a new function of J, say F (J), and thus:

Non-dimensional coefficients for thrust and torque are:

The other parameter of concern is the propeller efficiency which can be defined as the ratio of output to the input power. Thus:

170 2ti QN Kq 2 n

Thrust and torque coefficients and efficiency when plotted against advance coefficient produce plots as in Figure 9.13. Both thrust and

Figure 9.13 Thrust, torque and efficiency curves

Figure 9.13 Thrust, torque and efficiency curves torque coefficients decrease with increasing advance coefficient whereas efficiency rises to a maximum and then falls off steeply.

This format is good for presenting the data for a given propeller but not very useful for design purposes. In design the problem is usually to find the diameter and pitch of a propeller to provide the desired power at set revolutions and speed. The thrust power, PT, is the product of thrust and speed.

That is PT{N2/pVa5) = G (/), where G is a new function of J.

Taylor used U to denote thrust power and using seawater as the fluid, dropped p and took the square root of the left hand side of the above equation to give a coefficient Byj. He used a corresponding coefficient, Bp, for shaft power which he designated P. That is:

For a series of propellers in which the only parameter varied was pitch ratio, Taylor plotted Bu or Bp against pitch ratio in the form of contours for constant 6 values, <5 being the reciprocal of the advance coefficient. A typical plot is shown in Figure 9.14.

To use the plot the designer decides upon a value of revolutions for a given power and advance coefficient. This gives Bu or Bp. Erecting an ordinate at this value gives a choice of values of 6 from which the diameter is obtained. Associated with each diameter is a value of pitch ratio. For a given Bp the maximum efficiency that can be obtained is

that defined by the efficiency contour which is tangential to the ordinate at that Bp. In other words a line of maximum efficiency can be drawn through the points where the efficiency contours are vertical. Such a line is shown in Figure 9.14. The intersection of this line with the designer's Bp value establishes the pitch and diameter of the most efficient propeller.

Taylor used as units the horse power, speed in knots, Nin revolutions per minute and diameter in feet. With these units:

Keeping speed in knots and N in revolutions per minute, but putting diameter in metres and power in kilowatts:

The Taylor method of presentation is widely used for plotting model propeller data for design purposes.

### OPEN WATER TESTS

Open water tests of propellers are used in conjuncdon with tests behind models to determine the wake and reladve rotative efficiency. Also methodical propeller testing is carried out in a towing tank. The propeller is powered from the carriage through a streamlined housing. It is pushed along the tank with the propeller ahead of the housing so that the propeller is effectively in undisturbed water. Records of thrusi and torque are taken for a range of carriage speeds and propeller revolutions, that is advance coefficient. Such tests eliminate cavitation and provide data on propeller in uniform flow. This methodical series data can be used by the designer, making allowance for the actual flow conditions a specific design is likely to experience behind the hull it is to drive.

Va25 /V«Rev/min p «= shp (1 hp =» 76 kgm /s) d = diameter in feet fip — 8 diagram

Screw series B.4.40

H0 = pitch at blade tip

Pitch reduction at blade root = 20 %

Figure 9.15 Propeller diagram fip — 8 diagram

Screw series B.4.40

H0 = pitch at blade tip

Pitch reduction at blade root = 20 %

Va25 /V«Rev/min p «= shp (1 hp =» 76 kgm /s) d = diameter in feet

### Figure 9.15 Propeller diagram

There have been many methodical series. Those by Froude, Taylor, Gawn, Troost and van Lammeren are worthy of mention. The reader should refer to published data if it is wished to make use of these series. A typical plot for a four bladed propeller from Troost's series is presented in Figure 9.15.

### THE PROPELLER BEHIND THE SHIP

So far the resistance of the ship and the propeller performance have been treated in isolation. When the two are brought together there will be interaction effects.

Wake

The presence of the ship modifies the flow conditions in which the propeller works. The water locally will have a velocity relative to the ship and due to this wake, as it is called, the average speed of advance of the propeller through the local water will differ from the ship speed. The wake comprises three main elements:

(1) The velocity of the water as it passes round the hull varies, being less than average at the ends.

(2) Due to viscous effects the hull drags a volume of water along with it creating a boundary layer.

(3) The water particles in the waves created by the passage of the ship move in circular orbits.

The first two of these will reduce the velocity of flow into the propeller. The last will reduce or increase the velocity depending upon whether there is a crest or trough at the propeller position. If the net result is that the water is moving in the same direction as the ship the wake is said to be positive. This is the case for most ships but for high speed ships, with a large wave-making component in the wake, it can become negative. The wake will vary across the propeller disc area, being higher close to the hull or behind a structural element such as a shaft bracket arm. Thus the blades operate in a changing velocity field as the propeller rotates leading to a variable angle of incidence. The pitch cannot be constantly varied to optimize the angle and an average value has to be chosen. That is the design of each blade section is based on the mean wake at any radius.

Model tests in a towing tank can be used to study the wake but it must be remembered that the boundary layer thickness will be less relatively in the ship. Model data has to be modified to take account of full-scale measurements as discussed later.

In preliminary propeller design, before the detailed wake pattern is known, an average speed of flow over the whole disc is taken. This is usually expressed as a fraction of the speed of advance of the propeller or the ship speed. It is termed the wake fraction or the wake factor. Froude used the speed of advance and Taylor the ship speed in deriving the wake fraction, so that if the difference in ship and local water speed is

These are merely two ways of defining the same phenomenon. Generally the wake fraction has been found to be little affected by ship speed although for ships where the wave-making component of the wake is large there will be some speed effect due to the changing wave pattern with speed. The full-scale towing trials of HMS Penelope indicated no significant scale effect on the wake.6

The wake will vary with the after end shape and the relative propeller position. The wake fraction can be expected to be higher for a single screw ship than for twin screws. In the former the Taylor wake fraction may be as high as 0.25 to 0.30.

### Relative rotative efficiency

The wake fraction was based on the average wake velocity across the propeller disc. As has been explained, the flow varies over the disc and in general will be at an angle to the shaft line. The propeller operating in these flow conditions will have a different efficiency to that it would have if operating in uniform flow. The ratio of the two efficiencies is called the relative rotative efficiency. This ratio is usually close to unity and is often taken as such in design calculations.

Augment of resistance, thrust deduction

In the simple momentum theory of propeller action it was seen that the water velocity builds up ahead of the propeller disc. This causes a change in velocity of flow past the hull. The action of the propeller also modifies the pressure field at the stern. If a model is towed in a tank and a propeller is run behind it in the correct relative position, but run independently of the model, the resistance of the model is greater than that measured without the propeller. The propeller causes an augment in the resistance. The thrust, T, required from a propeller will be greater than the towrope resistance, R The propeller-hull interaction effect can be regarded as an augment of resistance or a reduction in thrust. This leads to two expressions of the same phenomenon.

Augment of resistance, a = -

and:

Thrust deduction factor, t - ——— Hull efficiency

Using the thrust deduction factor and Froude's notation:

Now TVa is the thrust power of the propeller and RVS is the effective power for driving the ship, with appendages, at V^. Thus:

Using Taylor's notation, PE = (PT) (1 - t)/{\ - u^).

In terms of augment of resistance (1 - t) can be replaced by 1/(1 -fa). The ratio of PE to PT is called the hull efficiency and for most ships is a litde greater than unity. This is because the propeller gains from the energy already imparted to the water by the hull. Augment and wake are functions of Reynolds' number as they arise from viscous effects. The variation between model and ship are usually ignored and and the error this introduces is corrected by applying a factor obtained from ship trials.

The factors augment, wake and relative rotative efficiency are collectively known as the hull efficiency elements.

Quasi-propulsive coefficient (QPC)

As already explained, this coefficient is obtained by dividing the product of the hull, propeller and relative rotative efficiencies by the appendage coefficient. If the overall propulsive coefficient is the ratio of the naked model effective power to the shaft power:

The propulsive coefficient = QPC X transmission efficiency.

The transmission efficiency can be taken1 as 0.98 for ships with machinery aft and 0.97 for ships with machinery amidships. The difference is due to the greater length of shafting in the latter.

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