Potential Flow, Waves and Seakeeping

Rigid body ship motions in waves

A rigid ship in waves has six rigid-body degrees of freedom: surge, sway and heave for translation, and roll, pitch and yaw for rotation. These are conventionally indexed 1 to 6 in that order. The loads that drive the motions, namely waves, wind and current, are imposed by the surrounding sea and are determined by the sea state and the external geometry of the hull, such as its size, shape and draught. They do not depend on the internal structural strength, which governs only how the structure responds to the loads.

For a linear response the motion obeys (M+A)\ddot{x} + B\dot{x} + Kx = F, where A, B and K are the added mass, damping and stiffness matrices. Symmetry about the centreline plane decouples the motions into two groups: surge, heave and pitch (modes 1, 3, 5) form one group, and sway, roll and yaw (modes 2, 4, 6) form the other. Coupling between the two groups vanishes, but coupling within a group does not. The added mass matrix is symmetric, A_{ij}=A_{ji}, by reciprocity in potential flow, so coupled terms such as A_{35}=A_{53} are equal. Diagonal terms such as A_{33} (kg) and A_{55} (kg\cdotm^2) have different units and are generally unequal.

If the roll reference point moves from O to a point P at height z_P above O, the roll added mass changes by the sway-roll coupling term. With the clockwise-positive roll convention, the roll added mass about P is a_{44}-a_{24}z_P. The sway added mass a_{22} does not enter the roll coefficient.

A hydrostatic restoring force or moment exists only when a displacement changes the balance between buoyancy and weight. Heave gives a restoring force \rho g A_w z, where A_w is the waterplane area, and pitch gives a restoring moment governed by the longitudinal metacentric height. Surge and sway involve horizontal translation and produce no hydrostatic restoring.

With the Fourier convention F(\omega)=\int f(t)e^{-i\omega t}\,dt, differentiation in time becomes multiplication: \ddot{\eta}\rightarrow -\omega^2\eta and \dot{\eta}\rightarrow i\omega\eta. The frequency-domain response is therefore \eta_3(\omega)=F_3(\omega)/[-\omega^2(M+A_{33})+i\omega B_{33}+K_{33}].

Elementary potential flows and superposition

Elementary potential flows, namely the uniform stream, the source or sink, the doublet and the free vortex, each satisfy Laplace's equation. Because the equation is linear, the sum of their velocity potentials is also a valid potential. This is the principle of superposition, and it is the basis for building complex flows from simple ones.

For a point source of strength \sigma at (a,0), the potential is \frac{\sigma}{2\pi}\ln\sqrt{(x-a)^2+y^2}. A sink of the same strength carries the opposite sign. A uniform stream plus a source and a sink gives a closed body shape, and the total potential is the sum of the three terms, with the sign of the stream term set by the flow direction.

• Doublet plus uniform stream — gives the ideal flow past a circular cylinder without circulation. Adding a free vortex at the centre introduces circulation while the cylinder surface remains a streamline.
• Source plus uniform stream — produces a stagnation point upstream of the source at a distance \sigma/(2\pi U) when the potential is \frac{\sigma}{2\pi}\ln r and U is the stream speed. At that point both velocity components vanish.

Only combinations that keep the body surface a streamline represent a closed body. Source and sink combinations that carry a net outflow through the body do not give a circular cylinder.

Flow past submerged bodies

Ideal flow past a circular cylinder without circulation is symmetric fore and aft. Adding circulation \Gamma with a free vortex produces a net lift on the cylinder. The lift per unit span is given by the Kutta-Joukowski theorem, L'=\rho U\Gamma, where \rho is the fluid density and U the free stream speed.

A rotating cylinder induces circulation through its surface speed. If the surface speed is v_s=\omega a for radius a, the circulation is \Gamma=2\pi a v_s=2\pi a^2\omega. Each stagnation point on the surface satisfies |\sin\theta_s|=\Gamma/(4\pi U a). Two stagnation points exist on the surface when \Gamma<4\pi U a, merge into one at the critical value \Gamma=4\pi U a, and do not exist when the circulation exceeds that value.

The lift on a section of a cylinder of length \ell is the lift per unit span multiplied by \ell, with the density entering in the same way. Keep the units consistent, since circulation has units of m^2/s.

For a non-lifting smooth body in steady, incompressible, irrotational flow, the potential solution is fore-aft symmetric and the integrated pressure force along the stream vanishes. This is d'Alembert's paradox: ideal potential flow predicts zero drag, and real drag arises from viscous effects.

Heave added mass and natural period

Heave motion of a floating body is governed by the effective mass M+A_{33} and the restoring stiffness \rho g A_w, where A_w is the waterplane area. The heave natural period is T_n=2\pi\sqrt{(M+A_{33})/(\rho g A_w)}. Heave added mass lengthens the period compared with the unassisted value 2\pi\sqrt{M/(\rho g A_w)}. When the added mass equals the displacement mass, the effective mass doubles and the period becomes \sqrt{2} times the unassisted value. Because displacement mass is \rho\nabla and stiffness is also proportional to \rho, the density cancels in this case.

Stiffness is set by the waterplane area, so structures with a small waterplane have low heave stiffness and long periods. A fixed jacket has no free heave motion and is the stiffest of the three types. A large-waterplane FPSO is stiffer than a semi-submersible whose columns have small waterplane areas. Hence the heave natural frequency is highest for the jacket and lowest for the semi-submersible.

Encounter frequency of waves with ship

When a ship moves relative to waves, the frequency that an observer on board measures differs from the true wave frequency. In deep water, the encounter frequency is \omega_e=\omega-\frac{\omega^2 U}{g}\cos\beta, where \omega is the wave frequency, U the ship speed and \beta the angle between the wave direction and the ship heading. In a head sea, \beta=180^\circ and \cos\beta=-1, so the encounter frequency is higher than the wave frequency.

The period counted on board is the encounter period T_e=2\pi/\omega_e. To recover the true wave period, invert the relation as a quadratic in \omega and choose the physically consistent positive root. Oblique seas use the same formula with the appropriate value of \cos\beta, which may be negative or positive depending on the heading angle.

• Head sea — waves approach from the bow, \cos\beta=-1, and the encounter frequency is increased.
• Following sea — waves approach from astern, and the encounter frequency is reduced at moderate speeds.

Response spectrum and motion variance

For a linear system excited by a stationary Gaussian sea, the response spectrum is S_R(\omega)=|H(\omega)|^2 S(\omega), where H(\omega) is the complex response amplitude operator (RAO) and S(\omega) is the wave spectrum. The squared magnitude is used because the RAO is a complex amplitude ratio and the spectrum represents energy per unit frequency.

The variance of the response equals the area under the response spectrum: \sigma^2=\int_0^\infty |H(\omega)|^2 S(\omega)\,d\omega. In practice the integral is evaluated numerically by multiplying the squared RAO by the spectral ordinates at each frequency and summing over the range where the spectrum has energy.