# Interaction of propeller and hull

The open-water properties of a propeller refer to the hydrodynamic performance of an isolated propeller in a uniform flow field. The ship resistance is generally considered the resistance of the isolated hull alone. The actual propeller works at the stern of the ship. The ship and the propeller form a system, and there is an interaction between the two. This interaction is manifested as the mutual influence between the velocity field formed by the hull and the velocity field formed by the propeller. Due to the influence of the hull on the propeller, the water flow velocity and distribution at the propeller disc are different from those in open water, and the water flow velocity distribution and pressure distribution around the hull are also different from those of the isolated hull due to the influence of the propeller. Therefore, the relative speed of the propeller and the water flow is not equal to the speed of the ship, and the thrust issued by the propeller is also different from the resistance received by the hull. At present, the method adopted when studying the propeller and the hull in engineering is to first study the influence of the ship on the propeller, then study the influence of the propeller on the ship, and then synthesize the two results in a simple form to obtain the final result.

**The effect of the hull on the propeller**

When the hull sails forward at a certain speed in the water, the nearby water will be affected by the hull and produce motion, which is represented by a current of water around the hull. This current is called the wake, and the wake makes the propeller’s relative velocity and the speed of the boat different from the current in its vicinity. The wake speed in the same direction as the ship speed is called positive wake, and the opposite is called negative wake. The wake mainly consists of three parts: friction wake, potential wake and wave wake. Friction wake is a following water current caused by the viscosity of water when the hull is in motion; if the ship moves forward for a certain distance, the bow must squeeze this section of water to both sides, and the stern will have a tendency to leave a section of water, so that the peripheral water is squeezed in from the bow and both sides; this kind of water flow that flows from the bow, through both sides to the stern with the movement of the hull is called the differential wake, and the effect of the differential wake is not very significant; the movement of the water accompanying the hull caused by the action of waves is called wave wake, which is generally small.

Therefore, the wake velocity u can be written as:

u=u_{p}+u_{f}+u_{w} (1-1)

In the formula, u_{p} is the axial average velocity of the differential wake at the paddle disc;u_{f} is the axial average velocity of the friction wake at the vegetable disc; u_{w} is the axial average velocity of the wave wake at the paddle disc.

Let the speed of the ship be v, the speed of Li relative to the water be v_{p}, and the water speed v_{p}<v flowing through the propeller due to the wake is defined as follows:

The wake speed is

U=v-v_{p} (1-2)

The wake coefficient is

ω=u/v=1-v_{p}/v (1-3)

then there are

J=h_{p}/D_{p}=v_{p}/nD_{p}=v(1-ω_{p})/nD_{p} (1-4)

The thrust generated by the propeller installed at the rear of the ship is larger than that of the propeller installed at the rear of the ship. The wake is a favorable factor for improving the thrust of the propeller. The propeller should be set at a position with a large wake as far as possible. When the ship gives energy to the water, a wake is generated, and the increase in thrust means that the ship recovers part of the energy.

**The effect of the propeller on the hull**

When the propeller is working behind the ship, due to its suction effect, the water flow speed in front of the propeller disk increases, resulting in a decrease in the pressure at the stern of the hull, which is equivalent to increasing the pressure difference between the bow and stern of the ship, thereby increasing the hull resistance. This additional drag on the hull caused by the propeller working behind the ship is called drag increase. If the total thrust from the propeller is P, a part of it is used to overcome the resistance P of the ship. (the resistance without the propeller), and the other part is to overcome the resistance increase ΔP, namely: P_{e}=P-ΔP. The thrust derating factor is defined as t=ΔP/P. Then there are:

P_{e}=P(1-t) (1-5)

The t of a single-hull ship can be determined by the following empirical formula:

t=C_{1}ω_{p} (1-6)

In the formula, ω_{p} is the wake coefficient; C_{1}=0.5~0.7 (a streamlined rudder is installed behind the propeller), C_{1}=0.7-0.9 (a square tail post and a double-plate rudder are installed behind the propeller), C_{1}=0.9~1.05 (plate rudder installed after propeller).

The t value of the double-paddle ship can use the following empirical formula:

t=0.25ω_{p}+0.14（Bossing）

t=0.70w_{p}+0.06（Propeller bracket）