25-Nav-A2 Hydrodynamics of Ships (I)_ Resistance and Propulsion · December 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — December 2019 — 16-Nav-A2 Hydrodynamics of Ships I: Resistance and Propulsion. Three-hour, closed-book exam; a data sheet, a propeller (Wageningen B4-55) chart and a Burrill cavitation chart are supplied. Format: Questions 1–7 are compulsory (attempt all seven), then one of Questions 8 or 9. All nine are solved below for completeness. Units follow the paper (mixed SI, with the historic Imperial-unit legend that duplicates on the supplied Burrill sheet noted where relevant).
Reference texts: Larsson & Raven, Ship Resistance and Flow (SNAME) — model-scale resistance testing, Froude/Reynolds scaling and the ITTC 1978 performance-prediction method; Lewis (ed.), Principles of Naval Architecture, Vol. II — Resistance, Propulsion and Vibration (SNAME) — propeller geometry, open-water B-series design and cavitation; Carlton, Marine Propellers and Propulsion (Butterworth-Heinemann) — Wageningen B-series charts and the Burrill back-cavitation criterion.
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.
(i) Pitch datum, pitch angle and pitch. The pitch datum line (also called the nose–tail line, or for some series the "pitch face" line) is the reference line on a blade section — normally the chord of the section, or a defined mean line for cambered sections — against which pitch is measured. The pitch angle $\varphi$ at radius $r$ is the angle this datum line makes with the plane perpendicular to the shaft axis (the "unrolled" transverse plane). The pitch $P$ is the axial distance the blade section would advance in one revolution if it moved along a true helix of that angle: $P=2\pi r\tan\varphi$. Because $\varphi$ generally varies with radius for a helicoidal blade, "the" propeller pitch usually means the pitch quoted at $0.7R$ (the reference radius used throughout this exam's data sheet).
(ii) Rake. Rake is the fore-and-aft displacement of the blade's generator line (the locus of the mid-chord, or section reference points, at each radius) from the plane perpendicular to the shaft axis, usually expressed as an angle or as a distance at the blade tip. Propellers are normally raked aft (toward the stern) to increase the clearance between the blade tips and the hull/aperture as the blade rotates past top-dead-centre, at some cost to propeller efficiency and increased bending stress at the root.
(iii) Sheet cavitation. Sheet cavitation is a large, coherent, attached vapour cavity that forms over a substantial area of the blade — typically the suction (back) side near the leading edge — when the local static pressure there falls below the vapour pressure of the water. It is distinguished from small-scale bubble or spot cavitation by being a continuous attached sheet rather than a cloud of discrete bubbles; the Burrill diagram used in Question 6 predicts the percentage of blade area affected by back sheet cavitation.
(iv) Camber and camber distribution. Camber is the asymmetry (curvature) of a blade section's mean line relative to its chord — the maximum perpendicular offset $f$ of the mean line above the chord is the camber, usually expressed as $f/c$. The camber distribution describes how this offset (and its chordwise location) varies along the chord at a given radius, and from root to tip across the blade; it is what gives the section net lift (thrust) even at zero geometric angle of attack.
(v) Tip vortex. At the blade tip the pressure difference between the face (pressure side) and back (suction side) cannot be sustained across the free tip edge, so flow rolls around the tip from face to back, forming a concentrated, trailing helical vortex core — the tip vortex — that is shed continuously and convects downstream in the propeller's slipstream. Its core pressure is often the lowest pressure anywhere in the flow field, making tip-vortex cavitation the earliest-inception form of propeller cavitation in many designs.
