Relative Rotative Efficiency (η_R)

Relative Rotative Efficiency (η_R) in Ship Propulsion

Relative rotative efficiency, denoted as η_R, represents the effect of hull-induced flow distortion on propeller torque and rotational performance. It compares the torque (or delivered power) required by a propeller operating behind the ship hull to that required in open-water conditions.

η_R captures rotational losses or gains caused by wake non-uniformity, hull-induced swirl, and altered blade loading, and forms a key component of the classical propulsion efficiency chain.

Definition and physical meaning

Relative rotative efficiency is defined as the ratio of open-water propeller torque to behind-ship propeller torque at the same operating condition:

η_R = Q₀ / Q_H

Where Q₀ is the propeller torque measured in open water and Q_H is the torque when operating behind the hull. Since delivered power is directly proportional to torque at a given rotational speed, η_R may also be interpreted as a power ratio.

Torque and delivered power interpretation

When propeller rotational speed is known, relative rotative efficiency can be expressed in terms of delivered power:

η_R = P_D,0 / P_D,H

This formulation is commonly used in model tests and full-scale trials, where shaft torque and rotational speed are measured directly. Differences in delivered power reflect how the hull wake modifies propeller loading.

Coefficient-based formulation

When torque coefficients are available instead of direct torque measurements, relative rotative efficiency can be computed using open-water and behind-ship torque coefficients:

η_R = K_Q,H / K_Q,0

This approach is widely applied in preliminary design and parametric studies, where propeller series data provides torque coefficients as a function of advance ratio.

Typical values and interpretation

• η_R < 1.0 — increased torque demand behind the hull.
• η_R ≈ 1.0 — neutral rotational interaction.
• η_R > 1.0 — favorable wake-induced rotation effects.
• η_R ≈ 0.95–1.05 — typical range for most merchant ships.

Values outside this range may indicate unusual wake fields, propeller–hull interaction issues, or inconsistencies in test data.

Limitations and correct usage

• η_R depends on hull wake characteristics and propeller location.
• It varies with loading condition, speed, and propeller operating point.
• Relative rotative efficiency alone does not describe overall propulsion efficiency.
• Consistent definitions of torque, power, and operating conditions are essential.

Related propulsion & power calculators

Relative rotative efficiency is one element of the complete propulsion efficiency chain:

• Propeller Efficiency (η₀) – open-water propeller performance.
• Hull Efficiency (η_H) – axial interaction between hull and propeller.
• Delivered Power (PD) – power required at the propeller.
• Brake Power (PB) – engine brake power including mechanical losses.

Tip: Relative rotative efficiency should always be evaluated together with hull efficiency and propeller efficiency to correctly assess propulsion system performance.