Jet Propulsion of Ships — Efficiency with Lateral (Right-Angle) Inlets For a ship using water-jet propulsion with inlet orifices arranged at right angles to the direction of motion, which expression gives the propulsive efficiency (η) in terms of ship speed V and jet speed relative to the ship V_j? Assume steady incompressible flow and ideal momentum exchange.

Introduction:
Jet propulsion of ships accelerates an incoming stream of water and ejects it astern to produce thrust. Propulsive efficiency compares useful power delivered to the hull (thrust * ship speed) with the rate of kinetic-energy input to the jet stream. When inlets are at right angles to the motion, the inflow has negligible streamwise momentum before acceleration, simplifying the efficiency relation.

Given Data / Assumptions:

• Ship speed relative to water: V.
• Jet exit speed relative to the ship: V_j (directed astern).
• Lateral (right-angle) inlets so the incoming water has near-zero axial momentum.
• Idealized momentum exchange and steady incompressible flow.

Concept / Approach:
The thrust T equals mass flow rate times the change in axial velocity. With right-angle inlets, the incoming axial component is approximately zero, and the outgoing axial velocity relative ground is V_j − V (backward). The propulsive efficiency is the ratio of useful power T * V to the jet power input in the slipstream. Under standard jet-propulsion analysis, this simplifies to η = 2V / (V + V_j), which shows that efficiency is highest when V is close to V_j (small velocity slip), but thrust then diminishes.

Step-by-Step Solution:
Let m_dot = rho * A * V_j (relative discharge through the pump and nozzle).Axial speed change (ground frame): 0 → (V_j − V) backward; thrust magnitude T = m_dot * (V_j − V).Useful power: P_use = T * V = m_dot * (V_j − V) * V.Jet power added: P_jet = (1/2) * m_dot * (V_j − V)^2 + (1/2) * m_dot * V^2 − approximately reduces to m_dot * (V_j^2 − V^2) / 2 under idealization.Propulsive efficiency reduces to η = 2V / (V + V_j).

Verification / Alternative check:
Limits: if V = V_j → η → 1 (perfect, but zero thrust). If V ≪ V_j → η ≈ 2V / V_j → low efficiency, matching practical experience.

Why Other Options Are Wrong:
V / V_j: ignores kinetic-energy balance; only a thrust ratio proxy.(V_j − V) / V_j: describes slip fraction, not efficiency.1 − (V / V_j)^2: energy ratio form that does not match momentum–energy analysis for jet propulsion.

Common Pitfalls:
Confusing air-jet and water-jet forms; the same ideal relation holds when inlet axial momentum is negligible.Using power-based slip factors without relating them to V and V_j.

Final Answer:
η = 2V / (V + V_j)