Resistance and Powering of Ships

1. Powering chain

Effective power is the power needed to tow the bare hull at a specified speed, without propeller action.

    flowchart LR
    A[Engine] --> B[BHP]
    B --> C[Reduction gear]
    C --> D[Shaft power SP]
    D --> E[Shafting bearings and seals]
    E --> F[Delivered power DP]
    F --> G[Propeller]
    G --> H[Thrust power TP]
    H --> I[Hull-propeller interaction]
    I --> J[Effective power EP]
  

The normal relationship is $BHP > SP > DP > TP > EP$, because power is lost through gears, bearings, seals, propeller action, friction, heat, vibration and sound.

• Brake horsepower (BHP): Engine output before the reduction gear.
• Shaft power (SP): Power available after the reduction gear.
• Delivered power (DP): Power delivered to the propeller.
• Thrust power (TP): Power generated by propeller thrust.
• Effective power (EP): Power required to overcome total hull resistance.

2. Power and efficiency formulas

Effective power is

$\mathrm{EP}=R_TV_S$

where $R_T$ is total resistance in newtons and $V_S$ is ship speed in metres per second.

The principal efficiencies are:

• Gear efficiency: $\eta_{\mathrm{gear}}=\frac{SP}{BHP}$
• Shaft transmission efficiency: $\eta_{\mathrm{shaft}}=\frac{DP}{SP}$
• Propeller efficiency: $\eta_{\mathrm{prop}}=\frac{TP}{DP}$
• Hull efficiency: $\eta_H=\frac{EP}{TP}$
• Overall propulsive efficiency: $\eta_P=\frac{EP}{SP}$

Therefore,

$\eta_P=\eta_{\mathrm{shaft}}\eta_{\mathrm{prop}}\eta_H$

and

$EP=SP\eta_P$

Thus, the required shaft power is

$SP=\frac{EP}{\eta_P}$

The overall power relationship is

$EP=BHP\eta_{\mathrm{gear}}\eta_{\mathrm{shaft}}\eta_{\mathrm{prop}}\eta_H$

Typical approximate values are:

Example: If $EP=23{,}500\ \mathrm{HP}$ and $\eta_P=0.68$,

$SP=\frac{23{,}500}{0.68}=34{,}559\ \mathrm{HP}$

3. Total hull resistance

Total hull resistance is the force opposing ship motion.

The basic resistance components are

$R_T=R_{AA}+R_W+R_V$

where:

• $R_{AA}$: Calm-air resistance on the above-water structure.
• $R_W$: Wave-making resistance.
• $R_V$: Viscous resistance, including frictional and form resistance.

Additional resistances may include appendage resistance, steering resistance, added resistance in waves and shallow-water resistance.

Main concepts

• At low speed, viscous resistance generally dominates.
• At higher speed, wave-making resistance rises rapidly.
• Air resistance is commonly about $4\%-8\%$ of total resistance.
• Appendage resistance may be significant, especially in naval vessels.
• Fouling, corrosion and rough paint increase resistance.
• Shallow water increases flow velocity beneath the hull, pressure changes, wave-making resistance and frictional resistance.

4. Resistance coefficients

The coefficient of total resistance is

$C_T=\frac{R_T}{\frac{1}{2}\rho S V_S^2}$

Therefore,

$R_T=C_T\left(\frac{1}{2}\rho S V_S^2\right)$

where:

• $\rho$: Water density.
• $S$: Wetted surface area.
• $V_S$: Ship speed.

The total coefficient may be expressed as

$C_T=C_V+C_W+C_A$

or, when correlation allowance is included,

$C_T=C_V+C_W+C_A+C_{A,\mathrm{corr}}$

where:

• $C_V$: Viscous-resistance coefficient.
• $C_W$: Wave-making-resistance coefficient.
• $C_A$: Air-resistance coefficient.
• $C_{A,\mathrm{corr}}$: Correlation allowance for full-scale effects.

5. Viscous resistance

Viscous resistance results from water viscosity and pressure changes around the hull.

The viscous coefficient is

$C_V=C_F+KC_F=(1+K)C_F$

where:

• $C_F$: Skin-friction coefficient.
• $K$: Form factor.
• $KC_F$: Viscous pressure or form-drag component.

The Reynolds number is

$Re=\frac{V_SL}{\nu}$

where $L$ is ship length and $\nu$ is kinematic viscosity.

A commonly used ITTC skin-friction relation is

$C_F=\frac{0.075}{(\log_{10}Re-2)^2}$

Approximate flow regimes over a flat plate are:

• Laminar flow: $Re<5\times10^5$.
• Turbulent flow: $Re>5\times10^5$.

Form factor

A fuller hull generally has a greater form factor and produces more pressure drag. A slender hull reduces pressure differences and eddy formation.

One empirical form-factor expression given in the notes is

$K=19\left(\frac{B}{L}\right)^2\left(\frac{T}{B}\right)^{0.5}\left(\frac{\nabla}{L^3}\right)^{0.5}$

where $B$ is beam, $T$ is draft and $\nabla$ is displacement volume.

Boundary-layer separation

    flowchart LR
    A[Flow along hull] --> B[Boundary layer develops]
    B --> C[Adverse pressure gradient]
    C --> D[Boundary-layer separation]
    D --> E[Wake and eddies]
    E --> F[Viscous pressure/form drag]
  

Boundary-layer separation causes a large wake and increases pressure drag. It can be reduced by using a slender, smoothly faired hull.

6. Froude number and wave resistance

The Froude number compares inertia forces with gravity forces:

$Fn=\frac{V_S}{\sqrt{gL}}$

where $g$ is gravitational acceleration and $L$ is characteristic ship length.

The speed-to-length ratio is

$\frac{V_S}{\sqrt{L}}$

but it is not dimensionless.

Wave resistance depends mainly on:

• Froude number.
• Beam-to-length ratio.
• Displacement.
• Hull shape.
• Ship length and speed.

Ship-generated waves consist mainly of:

• Transverse waves.
• Divergent waves from the bow and stern.

The approximate wavelength of the generated wave is

$\lambda_W=\frac{2\pi V_S^2}{g}$

The approximate hull-speed relationship is

$V_H=1.34\sqrt{L_{\mathrm{ft}}}\ \mathrm{knots}$

At hull speed, the transverse wavelength becomes approximately equal to ship length. Wave-making resistance then increases sharply.

• A hump occurs when bow and stern waves reinforce each other.
• A hollow occurs when they partially cancel each other.
• Increasing ship length raises hull speed.
• A bulbous bow can reduce bow-wave resistance through wave interaction.

7. Model testing and scaling

A towing-tank model requires geometric and dynamic similarity.

The scale factor is

$\lambda=\frac{L_S}{L_M}$

where subscripts $S$ and $M$ denote ship and model.

Scaling relationships are:

• Length: $L_S=\lambda L_M$
• Area: $S_S=\lambda^2S_M$
• Volume or displacement: $\nabla_S=\lambda^3\nabla_M$

For corresponding speeds, Froude’s law gives

$\frac{V_S}{\sqrt{L_S}}=\frac{V_M}{\sqrt{L_M}}$

Therefore,

$V_M=\frac{V_S}{\sqrt{\lambda}}$

and

$V_S=V_M\sqrt{\lambda}$

Use SI units for velocity:

$1\ \mathrm{knot}=0.5144\ \mathrm{m/s}$

For dynamically similar testing, the wave-resistance coefficient is transferred as

$C_{W,M}=C_{W,S}$

The model total-resistance coefficient is

$C_{T,M}=\frac{R_{T,M}}{\frac{1}{2}\rho_M S_MV_M^2}$

Hence,

$C_{W,M}=C_{T,M}-C_{V,M}$

For the full-scale ship,

$C_{T,S}=C_{V,S}+C_{W,S}+C_{A,S}+C_{A,\mathrm{corr}}$

Then,

$R_{T,S}=C_{T,S}\left(\frac{1}{2}\rho_SS_SV_S^2\right)$

Finally,

$EP_S=R_{T,S}V_S$

8. Propeller terminology

    flowchart LR
    A[Engine rotation] --> B[Propeller rotation]
    B --> C[Pressure difference over blades]
    C --> D[Water accelerated astern]
    D --> E[Forward thrust]
    E --> F[Ship advances]
  

Important parts and terms:

• Diameter $D$: Tip-to-tip distance.
• Hub: Central connection to the shaft.
• Blade root: Blade connection to the hub.
• Blade tip: Outermost part of the blade.
• Leading edge: First edge to meet incoming flow.
• Trailing edge: Rear edge of the blade.
• Pressure face: High-pressure side.
• Suction back: Low-pressure side.
• Pitch $P$: Theoretical forward distance per revolution in an ideal fluid.
• Propeller disc: Area swept by the blade tips.
• Projected area $A_P$: Blade area projected perpendicular to the shaft.
• Developed area: Actual blade driving-face area.
• Blade-area ratio: $BAR=\frac{\text{developed blade area excluding boss}}{\text{propeller disc area}}$
• Disc-area ratio: $DAR=\frac{\text{developed blade area including boss}}{\text{propeller disc area}}$

The pitch ratio is

$P/D=\frac{\text{pitch}}{\text{diameter}}$

Theoretical propeller speed is

$V_T=PN$

when $P$ is in metres per revolution and $N$ is in revolutions per second.

The propeller pitch angle is

$\tan\phi=\frac{P}{2\pi r}$

Types of pitch include:

• Fixed pitch: Blade pitch cannot be altered.
• Constant pitch: Pitch remains constant along the blade radius.
• Variable pitch: Pitch changes along the blade radius.
• Controllable pitch: Blade angle can be changed during operation.

9. Wake, advance speed and slip

The propeller operates in the wake produced by the ship.

Wake fraction is

$w=\frac{V_S-V_A}{V_S}$

Therefore, speed of advance is

$V_A=V_S(1-w)$

Approximate wake fraction:

$w\approx0.5C_B-0.05$

where $C_B$ is the block coefficient.

The theoretical propeller speed is

$V_T=PN$

Apparent slip is

$s_A=\frac{V_T-V_S}{V_T}$

Real slip is

$s_R=\frac{V_T-V_A}{V_T}$

Real slip is independent of current, whereas apparent slip is based on the ship’s speed through the water.

10. Thrust and propeller performance

For an idealised momentum model, the mass flow through the propeller disc per second is

$\dot m=\rho A_PPN$

The increase in water velocity is

$\Delta V=PN-V_A$

The thrust is

$T=\rho A_PPN(PN-V_A)$

Alternatively, if real slip $s$ is defined by $PN-V_A=sPN$,

$T=\rho A_P^2N^2s$

Thrust power is

$TP=TV_A$

Propeller efficiency is

$\eta_{\mathrm{prop}}=\frac{TP}{DP}=\frac{TV_A}{2\pi NQ}$

where $Q$ is propeller torque.

The thrust coefficient is

$K_T=\frac{T}{\rho n^2D^4}$

where $n$ is revolutions per second.

The thrust-loading coefficient is

$C_T=\frac{T}{\frac{1}{2}\rho A_OD^2?}$

Use the notation consistently with the propeller reference: the propeller disc area is $A_O$, and the loading coefficient is commonly written as

$C_T=\frac{T}{\frac{1}{2}\rho A_OV_A^2}$

A larger propeller diameter generally improves efficiency for a given thrust because it reduces disc loading, subject to draft, clearance and stern-form limitations.

11. Cavitation and ventilation

Cavitation occurs when local pressure on a propeller blade falls below the vapour pressure of water, producing vapour bubbles that subsequently collapse.

Effects include:

• Reduced propeller efficiency.
• Loss of thrust.
• Blade erosion.
• Vibration.
• Noise.
• Possible structural damage.

Cavitation becomes more likely when:

• Propeller thrust loading is high.
• RPM changes rapidly.
• Propeller diameter is too small.
• Blade surfaces are fouled, nicked or rough.
• Water pressure is low.

It can be reduced by:

• Increasing propeller diameter where practicable.
• Reducing thrust loading.
• Selecting suitable RPM and pitch.
• Maintaining smooth, clean blades.
• Changing RPM gradually.
• Operating submerged propellers at greater depth when possible.

Ventilation occurs when air or exhaust gas is drawn into a propeller operating close to the free surface. It reduces the water loading on the blades and produces effects similar to cavitation.

12. Practical resistance-reduction measures

• Keep the underwater hull clean and smooth.
• Prevent corrosion and marine fouling.
• Use an efficient, slender hull form.
• Operate below the resistance-hump speed where practical.
• Use a properly designed bulbous bow.
• Optimise propeller diameter, pitch and RPM.
• Minimise unnecessary appendages.
• Consider shallow-water effects when selecting operating speed.