Fluid Mechanics and Marine Hydrodynamics II: Potential Flow, Vortices, Lifting Surfaces, Added Mass, Model Testing and Surface Waves

Vorticity ω = ∇ × V is twice the local angular velocity of a fluid element. Circulation Γ = ∮V·dl around a closed curve equals, by Stokes’s theorem, the flux of vorticity through any surface spanning it, Γ = ∫∫ω·dA. Kelvin’s circulation theorem: in an inviscid, barotropic fluid under conservative body forces, the circulation around a closed curve moving with the fluid is constant in time. So a flow that starts from rest is irrotational and stays so — which is why the flow outside the boundary layer can be treated as potential, and why, when a foil starts moving and sheds a starting vortex, an equal and opposite circulation appears round the foil.

Irrotational flow has a velocity potential φ with V = ∇φ; if it is also incompressible, ∇·V = 0 gives Laplace’s equation ∇²φ = 0, which is linear, so solutions can be superposed. In two dimensions the stream function ψ (u = ∂ψ/∂y, v = −∂ψ/∂x) also satisfies Laplace’s equation; lines of constant ψ are streamlines, the difference of ψ between two streamlines is the flow rate between them, and streamlines cross equipotentials at right angles. For unsteady potential flow Euler’s equation integrates to the generalised Bernoulli equation ∂φ/∂t + p/ρ + ½|∇φ|² + gz = F(t), valid throughout the fluid and not just along a streamline — the form used for waves and for accelerating bodies.

Superposition builds bodies. A uniform stream plus a source gives the Rankine half-body, with a stagnation point at a distance m/(2πU) upstream of the source and a width tending to m/U far downstream. A stream plus a source and an equal sink gives a closed Rankine oval; letting the pair merge into a doublet gives the circular cylinder of radius a = √[κ/(2πU)], on whose surface the speed is 2U sin θ, the pressure coefficient Cp = 1 − 4 sin²θ, and the minimum Cp is −3 at the shoulders. In three dimensions, potential flow with rotational symmetry gives the sphere (a 3-D doublet in a stream), with surface speed 1.5U sin θ.

The pressure on the cylinder is symmetric front to back, so steady potential flow exerts no drag on any closed body — D’Alembert’s paradox, resolved by viscosity, which creates the boundary layer, separation and the wake. Adding a vortex of circulation Γ to the cylinder breaks the top–bottom symmetry: the stagnation points move to sin θ = −Γ/(4πUa) and the body feels a lift per unit span L′ = ρUΓ, perpendicular to the stream — the Kutta–Joukowski theorem, true for any two-dimensional body with circulation Γ. It explains the Magnus effect on a spinning cylinder (the Flettner rotor) and, with the Kutta condition, the lift of a foil.

Vorticity induces velocity exactly as current induces a magnetic field, so the Biot–Savart law carries over: an element dl of a vortex filament of strength Γ induces dV = (Γ/4π)(dl × r)/r³. An infinite straight filament induces a purely tangential speed Γ/(2πr) at distance r; a semi-infinite one, Γ/(4πr) at its end. Helmholtz’s theorems: the strength of a vortex filament is constant along it, a filament cannot end in the fluid (it closes on itself or ends on a boundary), and in an inviscid fluid it moves with the fluid.

Straight parallel vortex filaments move one another: two of equal and opposite strength a distance d apart translate together at Γ/(2πd), perpendicular to the line joining them; two of the same sign rotate about their centroid. A vortex sheet is a continuous distribution of vorticity of strength γ per unit length, across which the tangential velocity jumps by γ. Thin-aerofoil theory replaces the foil by a vortex sheet on its camber line, and the horseshoe vortex — a bound vortex along the span and two trailing tip vortices — is the model of a finite wing.

A foil in a stream generates lift because the Kutta condition — the flow leaves the sharp trailing edge smoothly — fixes the circulation. Thin-aerofoil theory gives the section lift coefficient C_l = 2π(α − α_L0) for small angles of attack α (in radians), with α_L0 = 0 for a symmetric section and negative for positive camber; the aerodynamic centre is at the quarter chord, where the pitching moment does not change with α, and for a symmetric section the centre of pressure is there too. The pressure distribution shows a suction peak near the leading edge on the upper (suction) side, and it is this peak that cavitates first.

On a wing of finite span the pressure difference drives flow round the tips, the circulation falls to zero at the tips, and the vorticity shed along the span forms a trailing vortex sheet that rolls up into tip vortices. The downwash it induces tilts the local flow, reducing the effective angle and producing induced drag. Prandtl’s lifting-line theory shows that an elliptic circulation distribution gives uniform downwash and the least induced drag, C_Di = C_L²/(πAR) (with a span-efficiency factor e < 1 otherwise, C_L²/(πeAR)), and reduces the lift slope to a = a₀/[1 + a₀/(πAR)], AR being the aspect ratio span²/area. The linearised theory of lifting surfaces extends this to low-aspect-ratio planforms such as rudders and propeller blades by distributing vortices over the whole surface.

Cavitation is the formation of vapour cavities where the local pressure falls to the vapour pressure p_v. It is governed by the cavitation number σ = (p∞ − p_v)/(½ρV²); a foil cavitates when −Cp,min exceeds σ, so high speed, shallow submergence (low p∞) and a sharp suction peak all promote it. Collapsing cavities cause erosion, noise and vibration, and extensive cavitation causes lift or thrust breakdown. Its forms on propellers — sheet, bubble, cloud, tip-vortex and hub-vortex — are taken up with propulsion.

A body accelerating through a fluid must accelerate some of the fluid with it, so it behaves as if its mass were increased by an added (hydrodynamic) mass; the extra force is −m_a (dU/dt), even in potential flow and even though a steadily moving body feels no drag. For a sphere the added mass is half the displaced mass, ½ρ(4/3)πa³; for a circular cylinder moving across its axis it is the displaced mass, ρπa² per unit length. Added mass depends on the direction of motion and on the free surface and depth, and it is the term that makes a ship’s heave and roll periods longer than its dry mass would suggest.

Slender-body theory exploits a body whose length greatly exceeds its beam: the flow at each cross-section is treated as two-dimensional, the section’s added mass is computed as if it were an infinite cylinder, and the forces follow from the rate at which the fluid momentum of the passing sections changes. A closed slender body in steady inviscid flow feels no net force but a destabilising (Munk) moment; the same cross-flow idea underlies strip theory in seakeeping and the prediction of manoeuvring forces.

Hydrodynamic model testing — in towing tanks, cavitation tunnels, manoeuvring and seakeeping basins — relies on scaling laws. Under Froude scaling with geometric scale λ = L_s/L_m and the same g, speeds and times scale as √λ, accelerations as 1, forces as λ³ (times the density ratio) and power as λ^3.5. Viscous effects scale with Re and cannot be matched at the same time, so friction is corrected separately; cavitation tests must also match σ, which is why cavitation tunnels reduce the ambient pressure.

Applying potential theory to the free surface with small amplitude gives linear (Airy) wave theory: a wave of height H, period T, wavenumber k = 2π/L and frequency ω = 2π/T obeys the dispersion relation ω² = gk tanh kh in depth h. In deep water (h > L/2, tanh kh → 1) ω² = gk, so the celerity is c = gT/(2π) and the length L = gT²/(2π) — a 10 s wave is about 156 m long and an 8 s wave travels at 12.5 m/s. In shallow water (h < L/20) c = √(gh), the same for all periods. Water particles move in circles in deep water, with radius decaying as e^kz below the surface, and in flattening ellipses in shallow water.

The wave energy per unit surface area is E = ρgH²/8, half kinetic and half potential, and it is carried not at c but at the group velocity c_g, which is c/2 in deep water and equal to c in shallow water; the energy flux (wave power per unit crest length) is E·c_g. Wave–body forces are split, in linear theory, into the Froude–Krylov force from the pressure of the undisturbed wave, the diffraction force from the body’s disturbance of it, and the radiation forces — added mass and wave damping — from the body’s own motion. For slender members of offshore structures, Morison’s equation gives the in-line force per unit length f = ½ρC_D D u|u| + ρC_M(πD²/4)(du/dt), a drag term plus an inertia term, with C_M = 1 + C_a.