# The working characteristics of the propeller and the resistance of the ship

## The working characteristics of the propeller

For a propeller with a certain geometry, its thrust coefficient K_{P}, drag torque coefficient K_{M} and efficiency η_{P} are only related to the advance speed ratio J, and the relationship between K_{P}, K_{M}, η_{P} and J is called the propeller characteristic curve. And because the performance of the propeller in question does not consider the influence of the hull, the curve is also called the open-water characteristic curve of the propeller, as shown in Figure 1. In the figure, because the K_{M} value is too small, a curve of 10 K_{M} is usually given. Neither K_{P} nor K_{M} is a straight line, and both are monotonically decreasing as J increases. It can generally be considered that K_{P} and K_{M} are approximately parabolas, and are expressed as

The coefficients K_{0}, K_{1}, K_{2}, etc. in the formula can be determined by curve fitting. If it is a given propeller, they are all constant coefficients.

From the operating characteristic curve in Figure 1, we can see that the K_{P}~J curve and the K_{M}~J curve are very similar in the first quadrant. In fact, the K_{P}~J curve and the K_{M}~J curve are relatively similar in the entire coordinate system. Figure 2 shows the K_{P}~J curve when J takes a wide range of values.

The curve from the upper left to the lower right in Figure 2 is the characteristic when the propeller speed n is forward rotation. The first quadrant corresponds to the state in which the propeller propels the ship forward, and the second quadrant corresponds to the state in which the ship turns to reverse operation, and the fourth quadrant part is imagined that a tugboat is dragging the ship forward, and the propeller becomes the working state of the water turbine. The curve from the upper right to the lower left in the figure is the characteristic when the propeller speed n is reversed. The fourth quadrant corresponds to the state in which the propeller pushes the ship to reverse, and the third quadrant corresponds to the state that the ship changes from reverse to forward, and the first quadrant corresponds to the working state in which it is assumed that a tugboat drags the ship backward, and the propeller becomes the working state of the turbine.

When the ship is actually sailing, it will be affected by various external factors, and its advance speed ratio J will change, so that the propeller thrust and drag torque associated with it will also change. It can be seen that starting from the working characteristics of the propeller, the working characteristics of the propeller in any working state of the ship can be simulated.

## Resistance of the ship

The ship will be resisted when sailing in the water, and the thrust generated by the propeller is used to overcome the resistance of the ship, so as to ensure its normal navigation. When the hull moves in the actual fluid, it will be subjected to the pressure perpendicular to the surface of the hull. This pressure is caused by waves and vortices; at the same time, the hull is subjected to the action of the tangential force of the water point along the surface of the hull, that is, the frictional resistance of the water. Therefore, the resistance of the ship when sailing includes vortex resistance, wave-making resistance and frictional resistance, and their sizes all increase with the increase of the ship’s speed. The relationship between the resistance and the speed is:

R=K_{r}V^{2} (1-3)

In the formula, R is the hull resistance; V is the speed; Kr is the resistance coefficient. When the operating conditions are constant, Kr is a constant, and when the operating conditions change, Kr also changes.

Usually, the actual ship sailing resistance curve is used, the curve value is stored in the database, and the resistance value corresponding to the ship speed at a certain moment is obtained by numerical interpolation algorithm.