Ship Resistance, Propulsion and Maneuvering

Propulsive coefficients and effective power

Effective power P_E is the useful power needed to tow the hull at speed V against the total resistance R_T, so P_E = R_T V. Power flows from the engine to the ship through a chain of losses: brake power P_B at the engine output, shaft power P_S after the gearbox, delivered power P_D at the propeller, thrust power P_T = TV_A, and finally effective power P_E. Each transmission efficiency is the ratio of two successive levels in this chain, and each is named after the loss it represents.

• Gear efficiency — ratio of shaft power to brake power, \eta_G = P_S/P_B.
• Shaft efficiency — ratio of delivered power to shaft power, \eta_S = P_D/P_S.
• Propeller behind-hull efficiency — ratio of thrust power to delivered power, \eta_B = P_T/P_D.
• Hull efficiency — ratio of effective power to thrust power, \eta_H = P_E/P_T = (1-t)/(1-w).

The quasi-propulsive coefficient is \eta_D = P_E/P_D = \eta_H\,\eta_0\,\eta_R, where \eta_0 is the open water efficiency and \eta_R is the relative rotative efficiency. Hull efficiency is not bounded by 1, because it depends on the thrust deduction t and the wake fraction w. When w > t it exceeds unity. For example, if t = 0.20 and w = 0.30, then \eta_H = 0.80/0.70 \approx 1.14.

The wake fraction is w = 1 - V_A/V, where V_A is the speed of advance of the open-water propeller that reproduces the behaviour of the propeller behind the hull. If the thrust identity and the torque identity give different advance speeds, the two wake fractions w_T and w_Q differ, and their mean is taken as the working value. Finally, if \eta_D = 0.65 and P_E = 1.2 MW, then P_D = 1.2/0.65 \approx 1.85 MW, and with a shaft efficiency of 0.97 the brake power is about 1.90 MW.

Froude scaling of ship and propeller models

Resistance tests in a towing tank follow Froude similarity, in which the model and the ship have the same Froude number Fr = V/\sqrt{gL}. With the scale ratio \lambda = L_s/L_m, the speed scales with \sqrt{\lambda}, the time scales with \sqrt{\lambda}, the area scales with \lambda^2, and the force scales with \lambda^3 when the same fluid density and gravity apply. The model speed is therefore V_m = V_s/\sqrt{\lambda}.

The residuary (wave-making) resistance coefficient C_R is the same for model and ship at corresponding speeds, so the ship's residuary resistance is obtained as R_R = C_R\cdot\frac{1}{2}\rho S V^2, using the ship's wetted area S and speed. Frictional resistance cannot be scaled in the same way, since the Reynolds number is not matched. Under Froude and Reynolds similarity together, the kinematic viscosity ratio must satisfy \nu_m/\nu_p = \lambda^{-3/2}, which is impossible in ordinary practice, so a frictional correction line is used.

For open water propeller tests, the advance coefficient J = V_A/(nD) must be identical for model and prototype. Since V \propto \sqrt{gD}, the rotational speed scales as n \propto D^{-1/2}.

Example: for a ship at 8 m/s and \lambda = 16, the model speed is 2 m/s. A model force of 50 N corresponds to a ship force of 50\times16^3 = 2.05\times10^5 N.

Straight-line stability in maneuvering

For a ship with fixed rudder, the linear sway-yaw equations can be written as \mathbf{M}\dot{\mathbf{x}} = \mathbf{N}\mathbf{x} with \mathbf{x} = [v\;\;r]^T. Stability of the straight-line motion is determined by the eigenvalues of \mathbf{M}^{-1}\mathbf{N}. If both eigenvalues are real and negative, disturbances in sway velocity and yaw rate decay monotonically, and the ship returns to a straight course. This property is straight line stability.

The usual non-dimensional test is the criterion C' = Y_v'(N_r' - m'x_G') - (Y_r' - m')N_v', where m' is the non-dimensional mass and x_G' is the centre of gravity position. A positive C' indicates stability and a negative C' indicates instability.

The following terms are easy to confuse:

• Straight line stability — return to a straight course after a small disturbance, with real negative eigenvalues.
• Directional stability — a property of the heading response, which is not the same as the straight-line test.
• Positional stability — a property of the lateral position of the ship, which is not determined by the eigenvalue signs alone.

Complex eigenvalues with negative real parts give damped oscillatory decay, and a zero eigenvalue gives neutral stability.

Pure sway PMM test

In a planar motion mechanism (PMM) test, the model is forced to oscillate in the transverse direction while it is towed at a steady forward speed U. In a pure sway test the transverse position is y = y_{max}\sin\omega t, with \omega = 2\pi/T where T is the period. The transverse velocity has maximum v_{max} = \omega y_{max}, and the sway acceleration has maximum a_{max} = \omega^2 y_{max}.

The drift angle is defined by \tan\beta = v/U, so the maximum drift angle satisfies \tan\beta_{max} = \omega y_{max}/U. Inverting this relation gives the amplitude required to reach a given drift angle, y_{max} = U\tan\beta_{max}/\omega.

Example: with an amplitude of 0.3 m, a period of 6 s and a carriage speed of 1.5 m/s, the angular frequency is \omega = 1.047 rad/s, the maximum lateral velocity is 0.314 m/s, and the maximum drift angle is \arctan(0.314/1.5) \approx 11.8^\circ. The hydrodynamic sway derivatives are obtained from the measured forces and moments over these motions.

Open water propeller efficiency

The open water efficiency of a propeller is the ratio of thrust power to torque power in open water, \eta_0 = \dfrac{J}{2\pi}\dfrac{K_T}{K_Q}, where J = V_A/(nD), K_T = T/(\rho n^2 D^4) and K_Q = Q/(\rho n^2 D^5). Many propeller tables list 10K_Q rather than K_Q, so the tabulated value must be divided by 10 before the formula is used.

The axial momentum (actuator disc) theory gives the thrust as T = 2\rho A u(V+u), where u is the induced axial velocity at the disc, V is the advance speed and A is the disc area. The ideal efficiency is then \eta = V/(V+u), because the useful power TV is divided by the total power T(V+u). For example, with V = 3 m/s and u = 1 m/s, the efficiency is 3/4 = 75\%. Real propellers have lower efficiency because of viscous and rotational losses.

An open water efficiency of 0.74 is typical of a well-designed propeller at moderate advance ratios; for example, with J = 0.7, K_T = 0.2 and K_Q = 0.03, \eta_0 = 0.7\times0.2/(2\pi\times0.03) \approx 0.74.

Propeller slip ratios

The pitch speed of a propeller is Pn, where P is the pitch and n is the rotational speed in revolutions per second. Two slip ratios are used in practice:

• Apparent slip ratio — based on the ship speed V, given by s_a = (Pn - V)/Pn.
• Real slip ratio — based on the speed of advance V_A = V(1-w), given by s_r = (Pn - V_A)/Pn.

Because V_A < V when the wake fraction is positive, the real slip ratio is always larger than the apparent slip ratio. For example, with P = 6 m, n = 2 rev/s, V = 9 m/s and w = 0.2, the pitch speed is 12 m/s, the apparent slip is 0.25 and the real slip is (12-7.2)/12 = 0.40.

The real slip is used to find the speed of advance, and the behind-hull efficiency links the thrust to the delivered power through \eta_B = TV_A/P_D. Thus, once V_A is known from the real slip, the thrust follows directly from T = \eta_B P_D/V_A.

Shallow water effects on ship speed

A ship is in shallow water when the depth-to-draught ratio is small. In this condition the flow under the hull is accelerated, which lowers the local pressure beneath the hull. The resulting dynamic changes are called squat, which is a sinkage together with a change of trim.