Fluid Mechanics and Ship Dynamics
Fluid Mechanics and Ship Dynamics applies continuity, momentum, Bernoulli, boundary-layer theory and rigid-body dynamics to ship motions. GATE NM tests both classical fluid mechanics and seakeeping concepts such as added mass, damping, RAO and roll period.
Core rules
- Continuity: A₁V₁ = A₂V₂ for incompressible steady flow. Bernoulli along a streamline: p/ρg + V²/2g + z = constant for ideal flow.
- Momentum force on a control volume equals the rate of momentum out minus in; account for pressure and friction separately.
- Reynolds number Re = VL/ν determines laminar or turbulent boundary layers; Froude number Fr = V/√(gL) governs wave-making.
- Heave, pitch and roll are coupled; added mass and damping depend on frequency.
- Natural roll period T = 2πk/√(g GM), where k is the roll radius of gyration and GM is metacentric height.
- Response amplitude operator (RAO) = response amplitude/wave amplitude at a given frequency; resonance occurs near the natural frequency.
Worked example
A ship has GM = 1.0 m, roll radius of gyration k = 6 m and g = 9.81 m/s². T = 2π × 6/√(9.81 × 1.0) = 37.70/3.132 ≈ 12.0 s. The reasoning is dimensional: k in metres divided by √(m/s² × m) gives seconds. If GM is reduced to 0.5 m, the period rises by √2 to about 17.0 s, which is why a stiff ship rolls quickly and a tender ship rolls slowly.
Traps
Using GM in centimetres or forgetting g gives a period off by a large factor. Added mass increases the effective inertia and lengthens the period, so the formula above is a simplified rigid-body result. Bernoulli is invalid across a pump, propeller or shock. Do not confuse Froude scaling of waves with Reynolds scaling of friction. Resonance is not the same as maximum wave height; it is a frequency match.