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25-Nav-A2 Hydrodynamics of Ships (I)_ Resistance and Propulsion · Undated paper

Question 6 of 10: Blade-Element Force Vectors

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — May 2019 — 16-Nav-A2 Hydrodynamics of Ships I: Resistance and Propulsion. Three-hour, closed-book exam; a data sheet, a Wageningen B4-55 propeller chart and a Burrill cavitation chart are supplied. Format: Questions 1–8 are compulsory (attempt all eight), then one of Questions 9 or 10. All ten are solved below for completeness.

Reference texts: Larsson & Raven, Ship Resistance and Flow (SNAME) — model-scale resistance testing, Froude/Reynolds scaling, boundary-layer estimates and the ITTC 1978 performance-prediction method; Lewis (ed.), Principles of Naval Architecture, Vol. II — Resistance, Propulsion and Vibration (SNAME) — propeller geometry, wave-pattern interference, open-water B-series design and wake-induced blade loading; Carlton, Marine Propellers and Propulsion (Butterworth-Heinemann) — Wageningen B-series charts, the Burrill back-cavitation criterion and open-water model testing.

Question 6: Blade-Element Force Vectors (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A blade element at radius $r$, with resultant velocity $V_R$ (the vector sum of axial advance speed $V_A$ and tangential rotational speed $2\pi nr$) arriving at angle $\varphi$ to the plane of rotation, and effective angle of attack $\alpha_E$ between $V_R$ and the section's chord line.

Find. The elemental lift, drag, thrust and torque-producing force vectors, and the equations resolving lift/drag into the axial and tangential directions.

Approach. At any radius the blade section sees a resultant inflow velocity $V_R$ (vector sum of the axial advance speed $V_A$ and the tangential rotational speed $2\pi nr$) arriving at angle $\varphi$ to the plane of rotation; the hydrodynamic lift and drag are defined relative to $V_R$ rather than relative to the ship's axial direction, and are then resolved into the axial (thrust) and tangential (torque) directions using the angle $\varphi$ between $V_R$ and the plane of rotation. This is the same resultant-velocity picture used in Question 10, where a wake-induced change in $V_A$ changes $\varphi$ and hence the angle of attack and lift at each blade position.

plane of rotation (tangential, U = 2πnr) chord / pitch line V_R (resultant velocity) φ α_E L (lift, ⊥ V_R) D (drag, ∥ V_R) dT (thrust, axial = L cosφ − D sinφ) dF_Q (tangential = L sinφ + D cosφ)
Figure 3 — Blade-element force diagram at radius $r$. Lift $L$ acts perpendicular to the resultant velocity $V_R$, drag $D$ acts along $V_R$; resolving both into the axial and tangential directions gives the elemental thrust $dT$ and the elemental tangential (torque-producing) force $dF_Q$.
VectorDirectionResolved form
Lift $L$⊥ to $V_R$—
Drag $D$∥ to $V_R$—
Elemental thrust $dT$axial (shaft direction)$L\cos\varphi-D\sin\varphi$
Elemental tangential force $dF_Q$tangential (plane of rotation)$L\sin\varphi+D\cos\varphi$, with $dQ=r\,dF_Q$
  1. Resolve lift and drag relative to $V_R$. By definition, lift $L$ acts perpendicular to the resultant velocity $V_R$ and drag $D$ acts parallel to (in the direction of) $V_R$, both applied at the blade section.
  2. Resolve into thrust (axial) and torque-producing (tangential) components. With $\varphi$ the angle of $V_R$ to the plane of rotation, $$dT=L\cos\varphi-D\sin\varphi\quad\text{(axial direction, thrust)},$$ $$dF_Q=L\sin\varphi+D\cos\varphi\quad\text{(tangential direction, produces torque }dQ=r\,dF_Q\text{)}.$$ Both the lift and drag components contribute to the tangential force $dF_Q$ in the same (torque-absorbing) sense, whereas drag always subtracts from the thrust contributed by lift — this is why a high lift-to-drag ratio at each blade section, not lift alone, is what makes a propeller efficient.