3.1. Simplifying assumptions

In the model, we suppose the following.

1. (i) The incoming wave is a low amplitude, inviscid potential gravity wave in deep water – in the experiments the maximum wave slope is $\epsilon = ka = 0.23$; the deep-water assumption is not fully satisfied, with $\tanh (kH) \simeq 0.65$ for the largest wavelength.

2. (ii) Viscosity is negligible – the viscous stress on the floater is of the order $\eta a \omega / \delta _s$, where $\delta _s = (\nu /\omega )^{1/2}$ is the thickness of the Stokes boundary layer. This viscous stress is much smaller than the pressure variations in the wave, $p \sim \rho a \omega ^2 / k$, for $\nu k^2/\omega \ll 1$, which is well satisfied in the experiments: $\nu k^2/\omega < 2 \times 10^{-4}$.

3. (iii) Capillarity is negligible – the capillary length, $\ell _c = \sqrt {\rho g / \gamma } \simeq 2.5$ mm, is smaller than the characteristic floater size and wavelength in our experiments. We ignore the change in equilibrium immersion depth $\bar {h}$ by the capillary forces.

4. (iv) Steady streaming flows are negligible – the steady streaming flow is a nonlinear, Eulerian mean flow correction of order $O(\epsilon ^2)$ that comes along with the wave. This mean flow can affect reorientation, but only if it is inhomogenous at the scale of the floater. Since steady streaming flows typically vary spatially on the scale $k^{-1}$, we expect a weak effect of streaming on the floater orientation for $kL \ll 1$.

5. (v) Wave scattering and emission are negligible – floaters are moving obstacles that scatter the incoming wave and that also emit waves. We ignore these flow modifications in our model, an approximation that is known as the Froude–Krylov approximation. This is only reasonable when the floater is small with respect to the wavelength and when differential motion is weak, i.e.

   (3.1a,b)\begin{equation} \delta= k L \ll 1 , \quad \frac{|| \boldsymbol{u} - \boldsymbol{v} ||}{|| \boldsymbol{u} ||} \ll 1, \end{equation}
   with $\boldsymbol {u}$ the fluid velocity and $\boldsymbol {v}$ the floater velocity. The Froude–Krylov approximation is not without physical consequences. By not considering diffraction, we filter out all radiative losses and added mass effects. One consequence is that free ‘bobbing’ oscillations of the floater around its equilibrium position are not damped in our model. Another consequence of the absence of dissipation is that, as for an undamped pendulum, the equilibrium states must be either unstable or marginally stable. Our model therefore predicts oscillations around the stable fixed points, rather than a convergence towards them as in the experiments.

3.2. Incoming wave

We define the incoming wave in the laboratory frame of reference $(O,x,y,z)$, with $(\boldsymbol {e}_x, \boldsymbol {e}_y, \boldsymbol {e}_z)$ the basis vectors (figure 5). By convention, the origin $O$ is on the equilibrium fluid surface. The wave propagates along $x$ and is $y$ invariant. Up to first order in wave magnitude, the potential wave solution on infinitely deep water is

(3.2a)\begin{equation} \zeta^{(1)}= a \sin (kx- \omega t ), \quad \left\{\begin{array}{@{}l@{}} u_x^{(1)} = a \omega {\rm e}^{kz} \sin (kx- \omega t), \\ u_z^{(1)} ={-} a \omega {\rm e}^{kz} \cos (kx- \omega t), \\ p^{(1)}= p_0 - \rho g z + \rho g a {\rm e}^{kz} \sin (kx - \omega t), \end{array} \right . \end{equation}

with $\zeta$ the surface elevation, $u_x,u_z,p$ the velocity components and pressure, $p_0$ the atmospheric pressure and $\omega = \sqrt {gk}$. Since we are interested in second-order effects in the floater motion, we also include second-order corrections in the incoming wave. At second order in wave magnitude, we have Stokes wave corrections of the surface and pressure but no extra flow,

(3.2b)\begin{equation} \zeta^{(2)} ={-} \frac{a^2 k}{2} {\cos (2 (kx- \omega t) ) } , \quad p^{(2)}={-} \frac{\rho g a^2 k}{2} {\rm e}^{2 k z} , \quad u_x^{(2)}= u_z^{(2)}=0 . \end{equation}

The pressure is slightly reduced everywhere under the wave and the surface locally steepens at wave crests and flattens at wave troughs.

Figure 5.

(a) Sketch of the floater and notations: laboratory frame $(O,x,y,z)$, moving floater frame $(C,\tilde {x},\tilde {y},\tilde {z})$ and Euler angles $\theta$ (pitch), $\phi$ (rall) and $\psi$ (yaw). In the simulation we numerically calculate the pressure force and moment on the submerged surface $S_{sub}$ using rectangular meshes on each face and a mask function $f$ that indicates whether the point on the face is submerged or not. (b) Free vertical oscillations of $z_c$ at the bobbing frequency $\sqrt {1/\beta \delta _z}$ in a numerical test case without an incoming wave.

In the following, we non-dimensionalise space, time, velocity and pressure using the scales

(3.3a–d)\begin{equation} [ \boldsymbol{r} ] = k^{{-}1} , \quad [t] = (gk)^{{-}1/2} , \quad [\boldsymbol{u}] = g^{1/2} k^{{-}1/2} , \quad [p] = \rho g k^{{-}1} . \end{equation}

In this unit system, the wave slope $\epsilon = ka \ll 1$ is the only remaining parameter that defines the wave and (3.2) become

(3.4a)$$\begin{gather} \zeta^{(1)} = \epsilon \sin (x- t ) , \quad \left\{\begin{array}{@{}l@{}} u_x^{(1)} = \epsilon {\rm e}^{z} \sin (x- t), \\ u_z^{(1)} ={-} \epsilon {\rm e}^{z} \cos (x- t), \\ p^{(1)}= p_0 - z + \epsilon {\rm e}^{z} \sin (x - t), \end{array} \right. \end{gather}$$

(3.4b)$$\begin{gather}\zeta^{(2)} ={-} \frac{\epsilon^2}{2} {\cos (2 (x- t) ) } , \quad p^{(2)}={-} \frac{\epsilon^2}{2} {\rm e}^{2 z} , \quad u_x^{(2)}= u_z^{(2)}=0. \end{gather}$$

3.3. Equations of motion for the floater

The floater is a rectangular parallelepiped with non-dimensional density $\beta$, non-dimensional length, width and height $( \delta _x , \delta _y, \delta _z) = (kL_x, kL_y,kL_z)$. We describe the instantaneous position of the centre of mass $C$ of the floater relative to $O$. The position vector is decomposed in the inertial, laboratory frame:

(3.5)\begin{equation} \boldsymbol{r}_c (t) = \boldsymbol{OC}(t) = x_c( t) \boldsymbol{e}_x +y_c( t) \boldsymbol{e}_y +z_c( t) \boldsymbol{e}_z. \end{equation}

To describe the orientation of the floater, we introduce a second reference frame, the non-inertial material frame $(C,\tilde {x},\tilde {y},\tilde {z})$, that is co-moving with the floater. By convention, the unit vectors $(\tilde {\boldsymbol {e}}_x (t), \tilde {\boldsymbol {e}}_y(t), \tilde {\boldsymbol {e}}_z(t))$ are aligned with the long, medium and short axes (figure 5). We introduce the three Euler angles, roll $\varphi (t)$, pitch $\theta (t)$ and yaw $\psi (t)$ that connect the laboratory frame to the moving floater frame. In our angle convention (see supplementary material available at https://doi.org/10.1017/jfm.2024.718) this transform is

(3.6a)\begin{equation} \left[\begin{array}{@{}c@{}} \boldsymbol{e}_x \\ \boldsymbol{e}_y \\ \boldsymbol{e}_z \end{array} \right] = \underbrace{\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{}} c_\psi c_\theta & \left( c_\psi s_\theta s_\varphi - s_\psi c_\varphi \right ) & \left ( c_\psi s_\theta c_\varphi + s_\psi s_\varphi \right ) \\ s_\psi c_\theta & \left (s_\psi s_\theta s_\varphi + c_\psi c_\varphi \right ) & \left ( s_\psi s_\theta c_\varphi - c_\psi s_\varphi \right ) \\ -s_\theta & c_\theta s_\varphi & c_\theta c_\varphi \end{array} \right]}_{R^T } \left[\begin{array}{@{}c@{}} \tilde{\boldsymbol{e}}_x \\ \tilde{\boldsymbol{e}}_y \\ \tilde{\boldsymbol{e}}_z \end{array} \right ] . \end{equation}

Here and further we denote in short $c_\psi = \cos \psi$, $s_\psi = \sin \psi$ and similarly for the other angles. This transform is easily inverted because $R$ is an orthogonal matrix, $R^{-1} = R^T$. Components of a vector $\boldsymbol {A}$ and the coordinates of the laboratory and floater frame are also connected by this matrix,

(3.6b)\begin{equation} \left [ \begin{array}{@{}c@{}} A_x \\ A_y \\ A_z \end{array} \right ] = R^T \left [ \begin{array}{@{}c@{}} \tilde{A}_x \\ \tilde{A}_y \\ \tilde{A}_z \end{array} \right ] , \quad \left [ \begin{array}{@{}c@{}} x - x_c \\ y -y_c \\ z - z_c \end{array} \right ] = R^T\left [ \begin{array}{@{}c@{}} \tilde{x} \\ \tilde{y} \\ \tilde{z} \end{array} \right ] . \end{equation}

In the absence of waves, $\epsilon =0$, the floater is in an equilibrium position, meaning perfectly levelled and with centre $C$ that is $\delta _z/2$ above the bottom face submerged at depth $\bar {h} = \beta \delta _z$. Hence, we have

(3.7a–d)\begin{equation} \bar{x}_{c} , \quad \bar{y}_{c} , \quad \bar{\psi} \quad \text{arbitrary} , \quad \bar{z}_c = \left ( \tfrac{1}{2} - \beta \right ) \delta_z , \quad \bar{\theta} = \bar{\varphi} = 0 \end{equation}

at equilibrium. Here and further, we will use overbars to label quantities that do not vary on the short time scale of the wave.

In the presence of waves, the floater will be displaced and by definition, it moves as a solid. An arbitrary point $\boldsymbol {r}$ of the floater has velocity

(3.8)\begin{equation} \boldsymbol{v} = \boldsymbol{v}_c (t) + \boldsymbol{\varOmega} (t) \times ( \boldsymbol{r} - \boldsymbol{r}_c (t) ), \end{equation}

with $\boldsymbol {v}_c (t) = \dot {\boldsymbol {r}}_c (t)$ the translation velocity of the centre of mass and $\boldsymbol {\varOmega } (t)$ the instantaneous rotation velocity. We decompose $\boldsymbol {v}_c (t)$ in the inertial, laboratory frame. The rotation vector is decomposed in the non-inertial, floater frame: $\boldsymbol {\varOmega } (t) = \tilde {\varOmega }_x (t) \tilde {\boldsymbol {e}}_x (t) + \tilde {\varOmega }_y(t) \tilde {\boldsymbol {e}}_y (t) + \tilde {\varOmega }_z(t) \tilde {\boldsymbol {e}}_z (t)$. The components of this rotation vector are kinematically linked to the time derivatives of the Euler angles (see supplementary material). We have

(3.9a,b)\begin{equation} \left [ \begin{array}{@{}c@{}} \dot{x}_c \\ \dot{y}_c \\ \dot{z}_c \end{array} \right ] = \left [ \begin{array}{c} {v}_{c,x} \\ {v}_{c,y} \\ {v}_{c,z} \end{array} \right ] , \quad \left [ \begin{array}{@{}c@{}} \dot{\varphi} \\ \dot{\theta} \\ \dot{\psi } \end{array} \right ] = \left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & \sin \varphi \tan \theta & \cos \varphi \tan \theta \\ 0 & \cos \varphi & - \sin \varphi \\ 0 & \dfrac{\sin \varphi }{\cos \theta} & \dfrac{\cos \varphi}{\cos \theta} \end{array} \right ] \left [ \begin{array}{@{}c@{}} \tilde{\varOmega}_x \\ \tilde{\varOmega}_y \\ \tilde{\varOmega}_z \end{array} \right ] . \end{equation}

From Newton's law and the angular momentum theorem, we have

(3.10a,b)\begin{equation} m \dot{\boldsymbol{v}}_c = \boldsymbol{F} , \quad \frac{{\rm d}}{{\rm d}t} \left ( \boldsymbol{I} \boldsymbol{\cdot} \boldsymbol{\varOmega} \right ) = \boldsymbol{K} , \end{equation}

with $m = \beta \delta _x \delta _y \delta _z$ the non-dimensional mass of the floater and $\boldsymbol {I}$ the non-dimensional inertia tensor. The principal moments of inertia of the rectangular parallelepiped are

(3.11a–c)\begin{equation} \tilde{I}_{xx} = \frac{m (\delta_y^2 + \delta_z^2 )}{12} , \quad \tilde{I}_{yy} = \frac{m (\delta_z^2 + \delta_x^2 )}{12} , \quad \tilde{I}_{zz} = \frac{m (\delta_x^2 + \delta_y^2 )}{12}. \end{equation}

The floater is subject to its weight and to pressure forces and moments. In non-dimensional form, we have

(3.12a,b)\begin{equation} \boldsymbol{F}= \int_{S_{sub}} ( p -p_0) \, {\rm d} \boldsymbol{S} - m \boldsymbol{e}_z , \quad \boldsymbol{K} = \int_{S_{sub}} ( \boldsymbol{r} - \boldsymbol{r}_c) \times (p -p_0) \, {\rm d} \boldsymbol{S} , \end{equation}

with ${\rm d}\boldsymbol {S}$ orientated towards the floater. In the Froude–Krylov approximation we use the incoming wave pressure $p$ and surface elevation defined in (3.4). Pressure and surface corrections due to scattered and emitted waves are entirely ignored. The submerged surface $S_{sub}$ corresponds to the part of the floater surface that has $z< \zeta$. We decompose Newton's law in the inertial laboratory frame and the angular momentum equation in the non-inertial floater frame. Using the fact that the material frame rotates along with the floater, i.e. that $\dot {\tilde {\boldsymbol {e}}}_i (t) = \boldsymbol {\varOmega }(t) \times \tilde {\boldsymbol {e}}_i (t)$, we obtain

(3.13a,b)\begin{equation} \left [ \begin{array}{@{}c@{}} m\dot{v}_{x,c} \\ m\dot{v}_{y,c} \\ m \dot{v}_{z,c} \end{array} \right ] = \left [ \begin{array}{@{}c@{}} F_{x} \\ F_{y}\\ F_{z} \end{array} \right ] , \quad \left [ \begin{array}{@{}c@{}} \tilde{I}_{xx} \dot{\tilde{\varOmega}}_x \\ \tilde{I}_{yy} \dot{\tilde{\varOmega}}_y \\ \tilde{I}_{zz} \dot{\tilde{\varOmega}}_z \end{array} \right ] = \left [ \begin{array}{@{}c@{}} {\tilde{K}}_{x} + ( \tilde{I}_{yy} - \tilde{I}_{zz} ) \tilde{\varOmega}_y {\tilde{\varOmega}}_z \\ {\tilde{K}}_{y} + ( \tilde{I}_{zz} - \tilde{I}_{xx} ) \tilde{\varOmega}_z {\tilde{\varOmega}}_x \\ {\tilde{K}}_{z} + ( \tilde{I}_{xx} - \tilde{I}_{yy} ) \tilde{\varOmega}_x {\tilde{\varOmega}}_y \end{array} \right ] . \end{equation}

Equations (3.13a,b) combined with the kinematic relations (3.9a,b) define a first-order system of 12 differential equations. If we provide an initial state for the floater position and velocity, we can numerically integrate this system forward in time.

3.4. Numerical simulations

In our numerical code we use the standard Runge–Kutta fourth-order explicit numerical scheme. The numerical calculation of the surface integrals (3.12a,b) that define the instantaneous $\boldsymbol {F}$ and $\boldsymbol {K}$ is non-trivial because the submerged surface $S_{sub}$ varies in time. We use the following procedure to compute these integrals (see sketch of figure 5a). On each of the six faces of the floater surface, we define two-dimensional rectangular meshes that contain $\tilde {x},\tilde {y},\tilde {z}$ coordinates on those faces, with typically $100 \times 100$ points. To evaluate the force components ${F}_{x}, {F}_{y},{F}_{z}$ at a given time $t$, we use a loop that visits all six faces. On each face, we first calculate the lab-frame $({x},{y},{z})$ coordinates of the points on that face, using the coordinate transform (3.6) and the present position $x_c(t) , y_c(t) , z_c(t) , \varphi (t), \theta (t), \psi (t)$. With these lab-frame coordinates, we can evaluate the pressure $(p-p_0)$ on that face using (3.4), using either a first-order description $p =p^{(1)}$ or a second-order description, $p =p^{(1)}+ p^{(2)}$. To handle the fact that the faces can be totally, partially or not submerged, we also calculate an indicator function $f= (1+\tanh ((\zeta (x,t)-z)/l)/2$, with $l$ a length over which the interface is smoothed and $\zeta$ as in (3.4), using again either a first-order description $\zeta =\zeta ^{(1)}$ or a second-order description, $\zeta =\zeta ^{(1)}+ \zeta ^{(2)}$. This function $f$, equal to $1$ in the liquid and $0$ in the air, is a smooth numerical approximation of the Heaviside function. We then take the product $(p-p_0) f$ that is only non-zero on the submerged points of that face. Using a two-dimensional quadrature rule and the numerical values of $(p-p_0) f$ on the face, we can then compute the surface integral on that face. Each face gives a local contribution to the force in the direction of the local inward normal $d \boldsymbol {S}$, so after having visited all six faces, we obtain the components $\tilde {F}_{x}, \tilde {F}_{y},\tilde {F}_{z}$ in the floater frame. Using the transform (3.6) we obtain the force components ${F}_{x}, {F}_{y},{F}_{z}$ in the lab frame. The moments $\tilde {K}_{x}, \tilde {K}_{y},\tilde {K}_{z}$ are calculated similarly.

We have done several static and dynamical tests in the absence of waves ($\epsilon =0$). In the static tests we validated the calculation of $\boldsymbol {F}$ and $\boldsymbol {K}$ on floaters that were submerged and rotated to positions for which we could easily compute the force and moment analytically. In the dynamical tests we used the code to reproduce free oscillations of the floaters. When we release the floater slightly off its equilibrium position (3.7a–d), we expect free ‘bobbing’ oscillations in the vertical $z_c$ or angular $\theta, \varphi$ coordinates, of non-dimensional frequencies (Falnes & Kurniawan Reference Falnes and Kurniawan2020)

(3.14a–c)\begin{align} \omega_z =\sqrt{\frac{1}{\beta \delta_z}}, \quad \omega_\theta = \sqrt{\frac{1}{\beta \delta_z} \left (\! \frac{\delta_x^2 + 6 \beta (\beta-1) \delta_z^2 }{\delta_x^2 + \delta_z^2} \!\right )},\quad \omega_\varphi = \sqrt{\frac{1}{\beta \delta_z} \left (\!\frac{\delta_y^2 + 6 \beta (\beta-1) \delta_z^2 }{\delta_y^2 + \delta_z^2} \!\right )}. \end{align}

Figure 5(b) shows an example of the time series for free vertical oscillations in $z_c$, for a floater with $\beta =0.5$ released slightly above its equilibrium position $\bar {z}_c=0$. As illustrated in the figure, the oscillatory motion $z_c = A \cos (\omega _z t)$ is accurately reproduced by our code.

The free oscillations in $z_c, \theta, \varphi$ are useful to test the code but do not reflect the behaviour of the floaters in our experiment. Restricting to floaters much smaller than the wavelength, $\delta \ll 1$, implies that the bobbing frequencies (3.14a–c) are much larger than the incoming wave frequency ($\omega =1$ in non-dimensional units). Accordingly, such fast bobbing oscillations are never resonant and are expected to be rapidly damped in the experiments, either by viscous friction or radiation loss. Since no damping is included in our model, we need to minimize these parasitic bobbing excitations. We use for this the following strategy: at time $t=0$, we place the floater at its equilibrium position (3.7a–d) and gradually ramp up the wave amplitude in time, by replacing $\epsilon$ by $\epsilon (1 - \exp (-t/T) )$ in the definition of pressure and surface height (3.4). Practice shows that with $T = 15 \times 2 {\rm \pi}$, we can keep the fast bobbing oscillations of the floater motion at low amplitude while capturing the slow dynamics induced by the wave motion.

We now consider numerical solutions for the floater motion in the presence of an incoming wave. We focus on the yaw angle motion. In a first series of simulations, we use the first-order pressure field and surface elevation: $p = p^{(1)}$ and $\zeta =\zeta ^{(1)}$. In figure 6 we show the time series of the yaw angle $\psi (t)$ for the same experimental conditions used in figure 2: floater size $L_y = 10$ mm, $L_z=5$ mm and $L_x$ varying from $30$ to $120$ mm, density ratio $\beta =0.44$, wavelength $\lambda =0.29$ m, wave steepness $\epsilon = ka = 0.16$ and initial yaw angle $\psi _0 = 45^\circ$. After a transient of typically 15 wave periods given by the ramp in the wave amplitude, the yaw angle $\psi$ shows fast oscillations (of period $2{\rm \pi}$) that correspond to the back-and-forth motion of the floaters, superimposed to slow oscillations around either the longitudinal position $\psi =0^\circ$ for short floaters or the transverse position $\psi =90^\circ$ for long floaters. The fast oscillations were not visible in figure 2(a) where the yaw angle measurement was synchronized at the wave frequency. The slow oscillations reflect the marginal stability of the fixed points in this dissipationless model. Apart from these differences, the numerical curves show a transition from longitudinal to transverse preferential orientation, somewhere in between $L_x = 70$ mm and $80$ mm, in excellent agreement with the experimental transition estimated at $L_{xc} = 75$ mm for this set of floaters.

Figure 6.

Time series for yaw angle for numerically simulated floaters with $\beta =0.44$, $L_y = 10$ mm, $L_z=5$ mm and $L_x$ varying from $30$ to $120$ mm. The incoming wave has wavelength $\lambda =0.29$ m and $\epsilon = ka = 0.16$, as in the experiments of figure 2. The transition from a longitudinal to a transverse preferential orientation occurs near $L_{xc} \approx 75$ mm, in good agreement with the experiments.

We have done similar simulations with first-order pressure $p = p^{(1)}$ and surface elevation $\zeta =\zeta ^{(1)}$ for more than 500 floaters, with parameters varied in broad ranges $\beta \in [0.03,0.97], \delta _x \in [0.004,5.43],\delta _y \in [0.01, 0.36], \delta _z \in [0.007,0.29], \epsilon \in [0.01,0.24]$. We summarize the preferential floater orientation in figure 7 using the same representation $(F,\epsilon )$ as in the experiments (figure 3). The transition from longitudinal to transverse orientation is located at $F_c = 60$ with no dependence in $\epsilon$. Near the transition $F_c$, we observe some outlying data points. This is not entirely unexpected as a narrow interval of bistability exists near the transition, as we show below. At high $\epsilon$, higher-order effects, neglected in the theory, may also become important.

Figure 7.

Preferential state of orientation in the $(F,\epsilon )$ plane according to the numerical simulations. More than 500 numerical simulations have been performed for varying $\beta, \delta _x,\delta _y, \delta _z, \epsilon$, showing an excellent agreement with the theoretical prediction $F_c=60$.

The slow rotation towards a preferential state of orientation is a second-order effect. Hence, it is legitimate to ask whether we still get the same results when the second-order correction of the Stokes wave is included, when $p = p^{(1)}+p^{(2)}$ and $\zeta =\zeta ^{(1)}+\zeta ^{(2)}$ in the simulations. In figure 8 we compare the time series of $\psi (t)$ with the first-order approximation (a) and the second-order approximation (b) for the same parameters as in figure 6. Both figures are nearly the same, a weak difference is only observable near the transition. This indicates that second-order corrections $p^{(2)}$ and $\zeta ^{(2)}$ do not have a dominant impact on preferential orientation and may in fact be ignored. This will also be theoretically justified in the next section.

Figure 8.

Time series for the yaw angle $\psi (t)$ of numerically simulated floaters with varying length $L_x$ (see figure 6 for details of the floaters and the wave), with (a) first-order pressure $p^{(1)}$ and surface elevation $\zeta ^{(1)}$ and (b) second-order corrections $\zeta ^{(2)}$ and $p^{(2)}$ also included. Preferential orientation is not much influenced by the second-order part of the Stokes wave.

4.1. Small $\epsilon$ expansion and equations of motions

We expand the motion as a perturbative series in powers of $\epsilon$ and denote

(4.1a,b)\begin{equation} \left \{ \begin{array}{@{}l@{}} x_c = \bar{x}_{c} (\tau) + x_c' (t) + O (\epsilon^2), \\ y_c = \bar{y}_{c}, \\ z_c = \bar{z}_{c} + z_c' (t) + O (\epsilon^2), \end{array} \right . \quad \left \{ \begin{array}{@{}l@{}} \varphi = 0 + \varphi' (t) + O (\epsilon^2), \\ \theta = 0 + \theta' (t) + O (\epsilon^2), \\ \psi =\bar{\psi}(\tau) + \psi' (t)+ O (\epsilon^2) . \end{array} \right . \end{equation}

At leading order $O(1)$, we recognise the equilibrium positions (barred variables). We admit that horizontal positions $\bar {x}_{c} (\tau )$ and mean yaw angle $\bar {\psi } (\tau )$ can vary slowly, on long time scales $\tau$. The mean position $\bar {y}_{c}$ cannot change in our Froude–Krylov model and is of no further concern. At order $O(\epsilon )$, we perturb the mean position with first-order deviations (primes) in position $x_c' (t), z_c'(t)$ and angle $\psi '(t),\theta ' (t), \varphi (t)$. These variables carry the harmonic response of the floater to the incoming wave. The main objective is to compute the second-order mean yaw moment that leads to the evolution equation (1.1) for the slow motion of the yaw angle $\bar {\psi } (\tau )$. We do not describe the second-order slow drift of $\bar {x}_{c} (\tau )$.

The roll and pitch angles $\varphi,\theta$ are small and this allows us to linearise all the dependencies on these angles: $s_\theta \approx \theta ' + O(\epsilon ^3) , c_\theta \approx 1+ O(\epsilon ^2) , s_\varphi \approx \varphi ' + O(\epsilon ^3) , c_\varphi \approx 1 + O(\epsilon ^2)$. The transform formula (3.6) reduces to

(4.2)\begin{equation} \left [ \begin{array}{@{}c@{}} \boldsymbol{e}_x \\ \boldsymbol{e}_y \\ \boldsymbol{e}_z \end{array} \right ] = \underbrace{ \left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} c_\psi & - s_\psi & \left ( c_\psi \theta' + s_\psi \varphi' \right ) \\ s_\psi & c_\psi & \left ( s_\psi \theta' - c_\psi \varphi' \right ) \\ -\theta' & \varphi' & 1 \end{array} \right ] }_{\mathcal{R}^T} \left [ \begin{array}{@{}c@{}} \tilde{\boldsymbol{e}}_x \\ \tilde{\boldsymbol{e}}_y \\ \tilde{\boldsymbol{e}}_z \end{array} \right ] + O (\epsilon^2), \end{equation}

and the inverse transform is defined with the transposed matrix $\mathcal {R}$ up to $O(\epsilon ^2)$.

We write a preliminary version of the evolution equations to identify the force and torque components that need to be calculated. The linearised evolution equations for the first-order deviations $x_c', z_c' , \varphi ' , \theta '$ are

(4.3a–d)\begin{equation} m \ddot{x}_{c}' = F_x' , \quad m \ddot{z}_{c}' = F_z' , \quad \tilde{I}_{xx} \ddot{\varphi}' = \tilde{K}_x' , \quad \tilde{I}_{yy} \ddot{\theta}' = \tilde{K}_y' . \end{equation}

Here, the forces $F_x', F_y'$ and moments $\tilde {K}_x', \tilde {K}_y'$ contain the $O(\epsilon )$, oscillatory part of the total force and moment. In the evolution equation for the yaw angle $\psi$ we keep $O(\epsilon )$ and $O(\epsilon ^2)$ terms, since we describe both the rapid motion of $\psi '$ and the slow motion of $\bar {\psi }$. From (3.9a,b) and (3.13a,b) and using $\tilde {\varOmega }_x \approx \dot {\varphi }', \tilde {\varOmega }_y \approx \dot {\theta }'$ we get

(4.4a,b)\begin{equation} \tilde{\varOmega}_z = \dot{\psi} - \varphi' \, \dot{\theta}' + O (\epsilon^3) , \quad \tilde{I}_{zz} \dot{\tilde{\varOmega}}_z = {\tilde{K}}_{z} + ( \tilde{I}_{xx} - \tilde{I}_{yy} ) \dot{\varphi}' \dot{\theta}' \end{equation}

up to order $O(\epsilon ^2)$. Elimination of $\tilde {\varOmega }_z$ gives

(4.5)\begin{equation} \tilde{I}_{zz} \dot{\psi} = {\tilde{K}}_{z} + \frac{{\rm d}}{{\rm d}t} \left(\tilde{I}_{zz} \varphi' \, \dot{\theta}' \right ) + ( \tilde{I}_{xx} - \tilde{I}_{yy} ) \dot{\varphi}' \dot{\theta}' . \end{equation}

We now remark the following simplification. From the transform (4.2) and using $\tilde {K}_x = \tilde {K}_x' + O(\epsilon ^2)$, $\tilde {K}_y = \tilde {K}_y' + O(\epsilon ^2)$ and the evolution equations (4.3a–d), we get

(4.6)\begin{align} K_z& \approx{-} \theta' \tilde{K}_x' + \varphi' \tilde{K}_y' + {\tilde{K}}_{z} + O(\epsilon^2) \nonumber\\ & \approx{-} \theta' \tilde{I}_{xx} \ddot{\varphi}' + \varphi' \tilde{I}_{yy} \ddot{\theta}' + {\tilde{K}}_{z} + O(\epsilon^2) . \end{align}

We reorganise this equation to isolate the component ${\tilde {K}}_{z}$ and substitute this into (4.5):

(4.7)\begin{equation} \tilde{I}_{zz} \ddot{\psi} = K_z + \underbrace{\frac{{\rm d}}{{\rm d}t} \left ( \tilde{I}_{xx} \dot{\varphi}' \theta' - \tilde{I}_{yy} \varphi' \dot{\theta}' + \tilde{I}_{zz} \varphi' \, \dot{\theta}' \right)}_{ \text{second-order harmonics} \sim {\rm e}^{{\pm} {\rm i} 2 t}} + O (\epsilon^3 ). \end{equation}

On the right-hand side we find $K_z$, the vertical moment component in the laboratory frame, next to a term ${\rm d}/{\rm d}t (\ldots )$ that is a time derivative of $O(\epsilon ^2)$ products. This second-order term has a vanishing time average, so it cannot affect $\bar {\psi }$. Hence, we can derive the equations for both yaw angle variables $\bar {\psi }$ and $\psi '$ from the simpler balance $\tilde {I}_{zz} \ddot {\psi } \approx K_z$. Separating the yaw moment in a rapidly varying $O(\epsilon )$ part $K_z'$ and a mean yaw moment $\bar {K}_z$ of order $O(\epsilon ^2)$, we obtain

(4.8a,b)\begin{equation} \tilde{I}_{zz} \ddot{\psi}' = K_z' , \quad \tilde{I}_{zz} \ddot{\bar{\psi}} = \bar{K}_z \end{equation}

as preliminary evolution equations for the yaw motion.

4.2. Theoretical calculation of force and moment components

To calculate the required force and moment components, we use an alternative formulation of (3.12a,b) in terms of the flow field $\boldsymbol {u}$ and with volume integrals. We replace ${\rm d} \boldsymbol {S}= - {\rm d} \boldsymbol {S}_{ext}$ in (3.12a,b), so that the surface element of the submerged volume points towards the fluid. Then using the divergence theorem on the interior of $S_{sub}$ and Euler's equation, $\partial _t \boldsymbol {u} +( \boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {\nabla } )\boldsymbol {u} + \boldsymbol {e}_z = - \boldsymbol {\nabla } p$, we get

(4.9a)$$\begin{gather} \boldsymbol{F} = \int_{V_{sub}} ( \partial_t \boldsymbol{u} + ( \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{u} ) \, {\rm d}V + \left ( \int_{V_{sub}} \, {\rm d}V - m \right ) \boldsymbol{e}_z, \end{gather}$$

(4.9b)$$\begin{gather}\boldsymbol{K} = \int_{V_{sub}} ( \boldsymbol{r} - \boldsymbol{r}_c) \times ( \partial_t \boldsymbol{u} + ( \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{u} ) \, {\rm d} V + \int_{V_{sub}} ( \boldsymbol{r} - \boldsymbol{r}_c) \times \boldsymbol{e}_z \, {\rm d} V \end{gather}$$

in the Froude–Krylov approximation, with $V_{sub}$ the submerged volume. We now inject the flow velocity $\boldsymbol {u} = \boldsymbol {u}^{(1)} + O (\epsilon ^3)$ as defined by (3.4) and introduce two approximations of the submerged volume:

1. $V_{sub}^{(0)}$: equilibrium submerged volume

2. $V_{sub}^{(0+1)}$: following the floater through its first-order motion, with surface at $z=\zeta ^{(1)}$.

We can integrate over both these volumes and also introduce the notation

(4.10)\begin{equation} \int_{V_{sub}^{(1)}} ( \ldots ) \,{\rm d}V = \int_{V_{sub}^{(0+1)}} ( \ldots ) \,{\rm d}V - \int_{V_{sub}^{(0)}} ( \ldots ) \,{\rm d}V . \end{equation}

This isolates the part in the integrals that is entirely due to the wave or floater motion. At leading order $O(1)$, without flow, we have the Archimedes balance

(4.11a,b)\begin{equation} \int_{V_{sub}^{(0)}} \, {\rm d}V - m = 0 , \quad \int_{V_{sub}^{(0)}} ( \boldsymbol{r} - \boldsymbol{r}_c) \times \boldsymbol{e}_z \, {\rm d} V =\boldsymbol{0} . \end{equation}

These equations can be used to find the equilibrium state (3.7a–d). For the first-order forces and moments, we have

(4.12a)$$\begin{gather} \boldsymbol{F}' = \int_{V_{sub}^{(0)}} \partial_t \boldsymbol{u}^{(1)} \, {\rm d}V + \int_{V_{sub}^{(1)}} {\rm d}V \,\boldsymbol{e}_z, \end{gather}$$

(4.12b)$$\begin{gather}\boldsymbol{K}' = \int_{V_{sub}^{(0)}} ( \boldsymbol{r} - \boldsymbol{r}_c) \times \partial_t \boldsymbol{u}^{(1)} + \int_{V_{sub}^{(1)}} ( \boldsymbol{r} - \boldsymbol{r}_c) \times \boldsymbol{e}_z \, {\rm d}V. \end{gather}$$

For the mean yaw moment, we need a second-order approximation. Using an overbar to indicate the time average over the short time scale $t$, we have

(4.13)\begin{equation} \bar{K}_z ={-} \overline{\int_{V_{sub}^{(0+1)}} (y-y_c) \, \partial_t u_x^{(1)} \, {\rm d} V} - \overline{ \underbrace{\int_{V_{sub}^{(0)}} (y-y_c) \, (\boldsymbol{u}^{(1)}\boldsymbol{\cdot} \boldsymbol{\nabla} ) u_x^{(1)} \,{\rm d} V }_{ {=}0 } } + O (\epsilon^3). \end{equation}

The advective term vanishes on average because $(\boldsymbol {u}^{(1)}\boldsymbol {\cdot } \boldsymbol {\nabla } ) u_x^{(1)} \sim \sin 2(x-t)$. We remark that no second-order wave characteristics appear in this mean yaw moment formula. This explains why second-order wave characteristics are not so important in the problem of preferential orientation, as we have seen in figure 8.

The integrals in (4.12) and (4.13) cannot be exactly calculated, but we can obtain asymptotic approximations in the small floater limit $\delta \ll 1$. The general idea is to use the Taylor series to replace the flow field and surface elevation in the vicinity of the small floater with polynomial approximations that are more easily integrated. We only present the main methods, details are in the supplementary material.

We parametrise the submerged volume in floater frame coordinates:

(4.14a–c)\begin{equation} V_{sub} : \ \tilde{x} \in [-\delta_x/2, \delta_x/2] , \quad \tilde{y} \in [-\delta_y/2, \delta_y/2] , \quad \tilde{z} \in [-\delta_z/2, \tilde{\zeta} (\tilde{x},\tilde{y},t)] . \end{equation}

Here $\tilde {\zeta } (\tilde {x},\tilde {y},t)$ represents the height of the water surface in the $\tilde {z}$ direction, as seen from $C$. We implicitly suppose that the top face of the floater is never partially submerged ($\tilde {\zeta } < \delta _z/2)$ and that the bottom face is always totally submerged ($\tilde {\zeta } > -\delta _z/2)$. This is a sensible approximation for low $\epsilon$ and intermediate values of $\beta$. To find a polynomial approximation of $\tilde {\zeta }$, we write a Taylor expansion of $\zeta (x,t)$ around $x_c$:

(4.15)\begin{equation} \zeta(x,t) = \zeta_c + (x-x_c ) \partial_x \zeta_c + \frac{(x-x_c )^2}{2} \partial^2_{xx} \zeta_c + \frac{(x-x_c )^3}{6} \partial^3_{xxx} \zeta_c + \dots. \end{equation}

The index ${}_c$ is used to express that the field is evaluated at the centre of the floater $C$, e.g. $\zeta _c = \zeta (x_c,t) = \epsilon \sin (x_c -t)$, $\partial _x \zeta _c = \partial _x \zeta |_{x=x_c} = \epsilon \cos (x_c -t)$, etc. We inject this expansion in the equation $z = \zeta (x,t)$ that defines the free surface and we replace both $z \approx z_{c} + ( - \theta ' \tilde {x} + \varphi ' \tilde {y} + \tilde {z} )$ and $x-x_c \approx c_\psi \tilde {x} - s_\psi \tilde {y} + O(\epsilon )$, the leading-order parts of the coordinate transform formula (4.2). Reorganising this into $\tilde {z} = \tilde {\zeta }(\tilde {x},\tilde {y}, t)$ gives

(4.16)\begin{align} \tilde{\zeta}(\tilde{x},\tilde{y}, t) &={-}\bar{z}_{c} - z_c' + \zeta_c + ( \theta' + c_\psi \partial_x \zeta_c ) \tilde{x} + ( -\varphi' - s_\psi \partial_x \zeta_c ) \tilde{y} \nonumber\\ & \quad + \tfrac{1}{2} \left ( c_\psi \tilde{x} - s_\psi \tilde{y} \right )^2 \partial^2_{xx} \zeta_c + \tfrac{1}{6} \left ( c_\psi \tilde{x} - s_\psi \tilde{y} \right )^3 \partial^3_{xxx} \zeta_c + \, O (\epsilon^2, \epsilon \delta^4) . \end{align}

The free surface is indeed $-\bar {z}_{c}$ above $C$ at equilibrium. The wave causes $O(\epsilon )$ corrections that here are separated in a local uniform elevation, linear inclination and quadratic and cubic corrections. The integral over the equilibrium submerged volume is defined as

(4.17)\begin{equation} \int_{V_{sub}^{(0)}} ( \ldots ) \,{\rm d}V = \int_{-\delta_x /2}^{\delta_x/2} \int_{-\delta_y/2}^{\delta_y/2} \int_{-\delta_z/2}^{ - \bar{z}_c } ( \ldots )|_{( \bar{x}_c,\bar{z}_c, \bar{\psi})} \,{\rm d} \tilde{x} \,{\rm d} \tilde{y} \, {\rm d} \tilde{z} . \end{equation}

The notation $|_{(\bar {x}_c,\bar {z}_c, \bar {\psi })}$ suggests that we need to replace $x_c=\bar {x}_c, z_c=\bar {z}_c, \psi = \bar {\psi }$ in the integrand because we need to evaluate it at the equilibrium position. The integral over the perturbed submerged volume $V_{sub}^{(0+1)}$ is more complex: we need to integrate up until the surface deformed by the wave and we also need to evaluate the integrand at the first-order perturbed floater position $x_c=\bar {x}_c+x_c', z_c=\bar {z}_c+z_c', \psi = \bar {\psi }+ \psi '$. We have

(4.18)\begin{align} & \int_{V_{sub}^{(0+1)}} \left( \ldots \right) {\rm d}V \nonumber\\ & \quad \approx \int_{-\delta_x /2}^{\delta_x/2} \int_{-\delta_y/2}^{\delta_y/2} \int_{-\delta_z/2}^{ \tilde{\zeta}(\tilde{x},\tilde{y}, t) } \left.\left( \ldots \right) \right|_{(\bar{x}_c+x_c', \bar{z}_c+z_c', \bar{\psi}+\psi')} \,{\rm d} \tilde{x} \,{\rm d} \tilde{y} \, {\rm d} \tilde{z} \nonumber\\ & \quad \approx \underbrace{ \int_{-\delta_x /2}^{\delta_x/2} \int_{-\delta_y/2}^{\delta_y/2} \int_{-\delta_z/2}^{ - \bar{z}_c } \left (1 + x_c' \frac{\partial}{\partial x_c} + z_c' \frac{\partial}{\partial z_c} + \psi' \frac{\partial}{\partial \psi} \right ) \left.\left( \ldots \right) \right|_{(\bar{x}_c, \bar{z}_c, \bar{\psi})} \,{\rm d} \tilde{x} \,{\rm d} \tilde{y} \, {\rm d} \tilde{z} }_{\text{equilibrium and deviation due to motion } {x_c',z_c',\psi'}} \nonumber\\ & \qquad + \underbrace{ \int_{-\delta_x /2}^{\delta_x/2} \int_{-\delta_y/2}^{\delta_y/2} \left( \tilde{\zeta}(\tilde{x},\tilde{y},t) + \bar{z}_c \right ) \left.\left(\ldots \right)\right|_{(\bar{x}_c,\bar{z}_c, \bar{\psi})} \,{\rm d} \tilde{x} \,{\rm d} \tilde{y} }_{\text{deviation due to locally varying submersion}} \ + O (\epsilon^2) . \end{align}

This formula is correct up to order $O(\epsilon )$ and requires some explanations. First of all, we have split the $\tilde {z}$ integral into two parts, one over the equilibrium submersion interval $\tilde {z} \in [-\delta _z/2,-\bar {z}_c]$ and one over the $O(\epsilon )$ deviation $\tilde {z} \in [-\bar {z}_c, \tilde {\zeta }]$. In the first part, we use a Taylor expansion $(1+ x_c' \partial _{x_c} + \dots )$ to re-express everything at the equilibrium position $x_c=\bar {x}_c, z_c=\bar {z}_c, \psi = \bar {\psi }$. In the second part, we can simplify the integration over $\tilde {z}$ as the integration interval has a size $\tilde {\zeta }+ \bar {z}_c$ that is of order $O(\epsilon )$.

Our choice to parametrise the integrals in floater frame coordinates requires that we also express the integrands in floater frame coordinates. We have $\boldsymbol {r}-\boldsymbol {r}_c = \tilde {x} \tilde {\boldsymbol {e}}_x + \tilde {y} \tilde {\boldsymbol {e}}_y + \tilde {z} \tilde {\boldsymbol {e}}_z$ and the coordinate transform (4.2). We use Taylor expansions around the floater centre $\boldsymbol {r}_c$ and replace all fields with polynomial approximations. Denoting in short $\boldsymbol {a}= \partial _t \boldsymbol {u}^{(1)}$ the local fluid accceleration, we have

(4.19)\begin{align} \boldsymbol{a} &= \boldsymbol{a}_{c} + (x-x_c) \partial_x \boldsymbol{a}_{c} + (z-z_c) \partial_z \boldsymbol{a}_{c} \nonumber\\ & \quad + \tfrac{1}{2} (x-x_c)^2 \partial_{xx}^2 \boldsymbol{a}_{c} + \tfrac{1}{2} (z-z_c)^2 \partial^2_{zz} \boldsymbol{a}_{c} + (x-x_c) (z-z_c)\partial^2_{xz} \boldsymbol{a}_{c} + O (\epsilon \delta^3) . \end{align}

We substitute the transform $x-x_c = c_\psi \tilde {x} - s_\psi \tilde {y} + ( c_\psi \theta ' + s_\psi \varphi ' ) \tilde {z}$ and $z-z_c = - \theta ' \tilde {x} + \varphi ' \tilde {y} + \tilde {z}$, keeping the order $O(\epsilon )$ terms that are proportional to $\theta '$ and $\varphi '$. This produces a lengthy expression detailed in the supplementary material that can be summarized as

(4.20)\begin{equation} \boldsymbol{a} = \boldsymbol{a}^{(1)} + \boldsymbol{a}^{(2)} + O (\epsilon^3) . \end{equation}

Next to the leading $O(\epsilon )$ term $\boldsymbol {a}^{(1)}$ there is a $O(\epsilon ^2)$ deviation $\boldsymbol {a}^{(2)}$ that is not related to $\boldsymbol {u}^{(2)} = \boldsymbol {0}$, but rather due to the fact that angular oscillations $\theta ', \varphi '$ cause the floater to feel spatial variations of the flow speed.

Having explained the main mathematical methods, we can now write explicit formula for the first-order force and torque components

(4.21a)$$\begin{gather} F_x' = \int_{V_{sub}^{(0)}} a_x^{(1)} \,{\rm d}V, \end{gather}$$

(4.21b)$$\begin{gather}F_z' = \int_{V_{sub}^{(0)}} a_z^{(1)} \, {\rm d} V + \int_{V_{sub}^{(1)}} \,{\rm d} V, \end{gather}$$

(4.21c)$$\begin{gather}\tilde{K}'_x = \int_{V_{sub}^{(0)}} \left ( \tilde{y} a_z^{(1)} + s_\psi \tilde{z} a_x^{(1)} - \tilde{z} \varphi' \right ) {\rm d}V + \int_{V_{sub}^{(1)}} \tilde{y} \,{\rm d} V, \end{gather}$$

(4.21d)$$\begin{gather}\tilde{K}'_y = \int_{V_{sub}^{(0)}} \left ( - \tilde{x} a_z^{(1)} +c_\psi \tilde{z} a_x^{(1)} - \tilde{z} \theta' \right ) {\rm d}V + \int_{V_{sub}^{(1)}} (- \tilde{x} ) \,{\rm d}V, \end{gather}$$

(4.21e)$$\begin{gather}K'_z ={-} \int_{V_{sub}^{(0)}} \left ( s_\psi \tilde{x} + c_\psi \tilde{y} \right ) a_x^{(1)} \,{\rm d}V . \end{gather}$$

For the second-order, mean yaw moment, we have

(4.22)\begin{align} \bar{K}_z &={-} \overline{ \int_{V_{sub}^{(0)}} \left ( s_\psi \theta' - c_\psi \varphi' \right ) \tilde{z} a_x^{(1)} \,{\rm d} V } - \overline{ \int_{V_{sub}^{(0)}} \left ( s_\psi \tilde{x} + c_\psi \tilde{y} \right ) a_x^{(2)} \,{\rm d} V } \nonumber\\ & \quad - \overline{ \int_{V_{sub}^{(1)}} \left ( s_\psi \tilde{x} + c_\psi \tilde{y} \right ) a_x^{(1)} {\rm d}V. } \end{align}

The calculation of these integrals is long but straightforward and in the supplementary material we provide some details. One practical question that appears is: to which order in non-dimensional floater size $\delta$ do we need to push the Taylor series? Considering the equations of motion (4.3a–d) and (4.8a,b), and the fact that $m= O(\delta ^3)$ and $\boldsymbol {I}= O(\delta ^5)$, we have calculated

(4.23)\begin{equation} \left.\begin{gathered} F'_x, F'_z \quad \text{up to order} \ O(\epsilon \delta^4), \\ \tilde{K}'_x, \tilde{K}'_y, K_z' \quad \text{up to order} \ O(\epsilon \delta^6), \\ \bar{K}_z \quad\text{up to order} \ O(\epsilon^2 \delta^6). \end{gathered}\right\} \end{equation}

This is necessary to access to the leading, $O(\epsilon )$ description of $x_c', z_c', \varphi ', \theta ', \psi '$, $O(\epsilon ^2)$ description of $\bar {\psi }$, but also to floater shape-related corrections of respective orders $O(\epsilon \delta )$ and $O(\epsilon ^2 \delta )$. In the limit $\delta \ll 1$, these shape corrections seem much smaller but this is without considering that these $O(\epsilon \delta )$ and $O(\epsilon ^2 \delta )$ terms are actually taking the form of $O(\epsilon \delta _x^2/\beta \delta _z )$ and $O(\epsilon ^2\delta _x^2/\beta \delta _z )$ terms. For strongly elongated floaters, we can have

(4.24)\begin{equation} F = \frac{\delta_x^2}{\beta \delta_z} \gg 1 \end{equation}

and in that case, the shape-related corrections become larger than the so-called leading-order terms. This is precisely what happens when the floaters change their preferential orientation. In practice, we really need to push the calculation to the orders (4.23), but we find that this results in a very long formula. This motivated us to add one extra assumption. In the experiment, all the floaters are always thinner in the $z$ direction and this means that

(4.25)\begin{equation} \delta_x , \delta_y \gg \delta_z \end{equation}

is a fair assumption. By exploiting this information in the calculation of the forces and moments we can ignore (i) all $O(\delta _z)$ terms with respect to $O(1)$ terms, (ii) all $O(\delta _z^2)$ terms with respect to $O(\delta _x^2, \delta _y^2)$ terms. This greatly simplifies the resulting formula for the forces and moments and produces a physically relevant result. We can in fact even further simplify the model by taking into account that our floaters are also very elongated $\delta _x \gg \delta _y$ in our experiments, but this will be done at the end.

5.1. A simpler formula for the yaw moment

We recall that both the rapid and slow motion of yaw angle $\psi$ are controlled by

(5.1)\begin{equation} K_z ={-} \int_{V_{sub}} (y-y_c) a_x \,{\rm d} V, \end{equation}

with $a_x = - \epsilon e^z \cos (x-t)$ the horizontal acceleration of the flow. We have seen at the end of the previous section that the limit of strongly elongated floaters $\delta _x \gg \delta _y \gg \delta _z$ is adapted to capture the preferential orientation phenomenon, so let us exploit this information directly. If the floater is indeed thin (along $\tilde {z}$) and not wide (along $\tilde {y}$), we can ignore the $\tilde {y}$ and $\tilde {z}$ variations in this moment integral and simplify it to

(5.2)\begin{equation} K_z \approx{-} \int_{-\delta_x/2}^{+\delta_x/2} (y-y_c) a_x \delta_y \underbrace{ \left( \tilde{\zeta}+\frac{\delta_z}{2} \right )}_{\tilde{h} (\tilde{x},t)} \,{\rm d} \tilde{x} . \end{equation}

For this formula to make sense, we must express $(y-y_c)a_x$ in terms of floater frame coordinates and ignore the $\tilde {y}$ and $\tilde {z}$ variations. We also need to find the local submersion depth $\tilde {h} (\tilde {x},t) = \tilde {\zeta }+({\delta _z}/{2})$ along the long $\tilde {x}$ axis of the floater. From (4.2) and ignoring all dependencies along $\tilde {y}$ and $\tilde {z}$, we find that

(5.3a–c)\begin{equation} x \approx x_c + c_\psi \, \tilde{x} , \quad y \approx y_c + s_\psi \, \tilde{x}, \quad z \approx z_c - \theta' \tilde{x} \end{equation}

along the long axis of the floater. The lever arm is then $y-y_c \approx s_\psi \, \tilde {x}$. The dependence of the field $a_x$ on $\tilde {x}$ can be found by injecting this coordinate transform in the theoretical expression. Then, making use of a Taylor expansion we get

(5.4)\begin{align} a_x &\approx{-} \epsilon e^{z_c - \theta' \tilde{x} } \cos ( x_c + c_\psi \tilde{x} - t ) \nonumber\\ & \approx{-} \epsilon \left(1 + z_c - \theta' \tilde{x} \right ) \left ( \cos ( x_c - t ) - c_\psi \tilde{x} \sin ( x_c - t ) \right ) + \dots \end{align}

up to $O(\epsilon ^2)$. To find the local submersion depth $\tilde {h}(\tilde {x},t)$, we ignore the $\tilde {y}$ dependencies in the general definition of $\tilde {\zeta }$ in (4.16). We also use the fact that $-\bar {z}_{c} + (\delta _{z}/2) = \beta \delta _z = \bar {h}$ is the equilibrium submersion depth. In this way, we find that the local submersion depth $\tilde {h}(\tilde {x},t)$ along the floater long axis is

(5.5)\begin{align} \tilde{h}(\tilde{x}, t) \approx \bar{h} + (- z_c' + \epsilon \sin (x_c-t ) ) + ( \theta' +\epsilon c_\psi \cos (x_c-t ) \tilde{x} - \frac{\epsilon}{2} c_\psi^2 \tilde{x}^2 \sin (x_c-t) . \end{align}

The wave surface at the scale of the floater is here approximated by a second-order polynomial and this is sufficient in this simplified model. In this expression, we need to insert the first-order motion $z_c'$ and $\theta '$. We already know the first-order motion from the asymptotic theory, but let us show how we can alternatively find this motion using some simpler physical arguments. In the pressure force and moment, the buoyancy terms are the largest ones, much larger than the dynamical pressure terms. Hence, the floater is moving in such a way that it keeps its submerged volume nearly constant in time. This requires

(5.6)\begin{equation} \int_{V_{sub}} \, {\rm d}V \approx \delta_y \int_{-\delta_x/2}^{+\delta_x/2} \tilde{h} (\tilde{x},t) \,{\rm d} \tilde{x} \approx m \quad \Rightarrow \quad \frac{1}{\delta_x}\int_{-\delta_x/2}^{+\delta_x/2} \tilde{h}(\tilde{x}, t) \,{\rm d} \tilde{x} \approx \bar{h} . \end{equation}

Evaluating the integral using the expression (5.5), we get the result

(5.7)\begin{equation} z_c' \approx \epsilon \sin (x_c-t ) \left ( 1 - \frac{\delta_x^2}{24} c_\psi^2 \right ) . \end{equation}

This indeed corresponds to the expression of $z_c'$ that we have obtained from a more formal treatment of the equations of motion, in the limit $\delta _x \gg \delta _y$ (see (4.26b)). Interestingly, we also recover the shape-related $O(\epsilon \delta _x^2)$ correction that is quite crucial in the model. To explain the angular motion $\theta '$, we can use a similar argument. Due to buoyancy, the floater will rotate so as to keep the Archimedes torque zero

(5.8)\begin{equation} \int_{V_{sub}} ( \boldsymbol{r} - \boldsymbol{r}_c) \times \boldsymbol{e}_z \, {\rm d} V \approx \boldsymbol{0}\ \Rightarrow \int_{-\delta_x/2}^{+\delta_x/2} \tilde{x} \tilde{h}(\tilde{x}, t) \,{\rm d} \tilde{x} \approx 0 \ \Rightarrow \theta' \approx{-} \epsilon c_\psi \cos (x_c-t ) . \end{equation}

This approximation is sufficient, the extra $O(\epsilon \delta _x^2)$ correction in $\theta '$ of (4.26d) is not so important. Replacing these expressions of $z_c'$ and $\theta '$ into (5.5), we find the following approximation for the local shape of the waterline at the floater position:

(5.9)\begin{equation} \tilde{h}(\tilde{x}, t) \approx \bar{h} + \epsilon c_\psi^2 \left (\frac{\delta_x^2}{24} -\frac{\tilde{x}^2}{2} \right ) \sin (x_c-t) . \end{equation}

The waterline as seen from the floater centre always takes the shape of a parabola, symmetrical around $\tilde {x}=0$. Combining the elements together, we find a formula for the yaw moment $K_z$:

(5.10)\begin{align} K_z & \approx{-} \int_{-\delta_x/2}^{+\delta_x/2} \underbrace{\tilde{x} s_\psi }_{\text{local lever arm}} \, \underbrace{\left (- \epsilon \left(1 + z_c - \theta' \tilde{x} \right ) \left ( \cos ( x_c - t ) - c_\psi \tilde{x} \sin ( x_c - t ) \right ) \right ) }_{ \text{local force density } f_x} \nonumber\\ & \quad \times \, \underbrace{ \left( \bar{h} + \epsilon c_\psi^2 \left (\frac{\delta_x^2}{24} -\frac{\tilde{x}^2}{2} \right ) \sin (x_c-t) \right ) }_{\text{local submersion } \tilde{h}} \, \delta_y \, {\rm d}\tilde{x} . \end{align}

This integral contains all the physics that explains the preferential orientation. The moment $K_z$ is the result of the local force density $f_x$ that varies along the floater's long axis applied on the local level arm. This force density is weighted by the local submersion depth $\tilde {h}$ and this effect is thus more important as the floater is longer.

5.2. Evolution equation for $\bar {\psi }$

We now evaluate the yaw moment (5.10) up to $O(\epsilon ^2)$ and write the evolution equation $\tilde {I}_{zz} \ddot {\psi } = K_z$. Using $\tilde {I}_{zz} \approx m \delta _x^2/12$ for our elongated floater, we find that

(5.11)\begin{equation} \ddot{\psi} \approx{-} \epsilon (1+ z_c) s_\psi c_\psi \sin (x_c-t) - \epsilon \theta' s_\psi \cos (x_c- t) + \frac{\epsilon^2 \delta_x^2}{30 \beta \delta_z} s_\psi c_\psi^3 \sin^2 (x_c -t ) . \end{equation}

We now inject in this equation the decomposition $x_c = \bar {x}_c + x_c'$ and $z_c = \bar {z}_c + z_c'$. In the first, $O(\epsilon )$ term of (5.11), we use a Taylor expansion

(5.12)\begin{align} & - \epsilon (1+ \bar{z}_c + z_c') s_\psi c_\psi \sin (\bar{x}_c + x_c'-t) \nonumber\\ & \quad ={-} \epsilon (1+ \bar{z}_c) s_\psi c_\psi \sin (\bar{x}_c -t) - \epsilon z_c' s_\psi c_\psi \sin (\bar{x}_c -t) - \epsilon x_c' s_\psi c_\psi \cos (\bar{x}_c -t) + O (\epsilon^3). \end{align}

As the floater is small and thin, we can approximate $\bar {z}_c \approx 0$. The second and third terms of (5.11) are already of order $O(\epsilon ^2)$, so there we can use $\sin ^2 (x_c - t ) \approx \sin ^2(\bar {x}_c - t ) + O (\epsilon )$ and $\cos (x_c - t ) \approx \cos (\bar {x}_c - t ) + O (\epsilon )$. We then substitute the first-order deviations

(5.13a–c)\begin{equation} x_c' \approx \epsilon \cos (\bar{x}_c -t ) , \quad z_c' \approx \epsilon \sin (\bar{x}_c -t ) , \quad \theta' ={-} \epsilon c_\psi \cos (\bar{x}_c - t). \end{equation}

The expression of $x_c'$ can be found by integrating $\dot {x}'_c = u_x |_{z=0}$, which just means that, at lowest order, the floater translates as a fluid material particle on the surface. The correction of order $O(\epsilon \delta _x^2)$ in $z_c'$ (see (5.7)) is not needed here. After these reductions, we obtain

(5.14)\begin{equation} \ddot{\psi} \approx{-} \epsilon s_\psi c_\psi \sin (\bar{x}_c-t) + \epsilon^2 s_\psi c_\psi \left ({-}1 + \frac{\delta_x^2}{30 \beta \delta_z} c_\psi^2 \right ) \sin^2 (\bar{x}_c -t ) \end{equation}

as the evolution equation for $\psi$. We now inject the decomposition $\psi = \bar {\psi } + \psi '$ and use a Taylor expansion to replace

(5.15)\begin{equation} s_\psi c_\psi = \bar{s}_\psi \bar{c}_\psi + \psi' (\bar{c}_\psi^2 - \bar{s}_\psi^2 ) + O (\epsilon^2), \end{equation}

yielding

(5.16)\begin{align} \ddot{\bar{\psi}} + \ddot{\psi}' &\approx{-} \epsilon \bar{s}_{\psi} \bar{c}_{\psi} \sin (\bar{x}_c-t) \nonumber\\ &\quad - \epsilon (\bar{c}_{\psi}^2 - \bar{s}_\psi^2 ) \psi' \sin (\bar{x}_c-t) + \epsilon^2 \bar{s}_\psi \bar{c}_\psi \left ({-}1 + \frac{\delta_x^2}{30 \beta \delta_z} \bar{c}_\psi^2 \right ) \sin^2 (\bar{x}_c -t ). \end{align}

At order $O(\epsilon )$, we identify the equation for the fast yaw angle excursion $\ddot {\psi }' = - \epsilon \bar {s}_{\psi } \bar {c}_{\psi } \sin (\bar {x}_c-t)$, which has the solution

(5.17)\begin{equation} \psi' \approx \epsilon \bar{s}_\psi \bar{c}_\psi \sin (\bar{x}_c-t). \end{equation}

This expression is identical to that of the asymptotic model (4.26e) in the limit $\delta _x \gg \delta _y$. Injecting this expression of $\psi '$ back into the equation and taking the average over the short time-scale, we find the second order evolution equation for $\bar {\psi }$:

(5.18)\begin{equation} \ddot{\bar{\psi}} \approx{-}\epsilon^2 \bar{s}_\psi \bar{c}_\psi^3 \left( 1 - \underbrace{\frac{\delta_x^2}{60 \beta \delta_z} }_{F/F_c} \right). \end{equation}

The simplified model reproduces exactly the more formal asymptotic theory.

5.3. Physical origin of preferential orientation

The simplified model allows a better understanding of the physics that causes reorientation. Let us reconsider the moment $K_z$ in (5.10), expressed as the product of the local force density $f_x$ by the local level arm weighted by the local submersion depth $\tilde {h}$. Both quantities being rapidly oscillating in time, a non-zero product arises from a phase correlation between them, that we illustrate in figure 12. Here, we show the floater at two wave phases, $t={\rm \pi} /2$ and $t=3{\rm \pi} /2$ (wave crest and trough), and represent the force density $f_x$ as vector arrows; animations are available as supplementary materials. The parameters are $\beta =0.2$, $\delta _x =0.5, \delta _z = 0.01$ and $\epsilon =0.2$, corresponding to a non-dimensional number $F = \delta _x^2 / \beta \delta _z = 125$, larger than the critical value $F_c=60$: this floater will prefer the transverse position at late times. To illustrate the effect of the variation in submersion, we show in the second column (figure 12b,e) the force distribution weighted by the equilibrium, spatially uniform submersion depth $\bar {h}$, whereas in the third column (figure 12c,f) it is weighted by the true, varying depth $\tilde {h}$.

Figure 12.

Position, orientation and force distribution on the floater in wave crests and troughs. Here $\delta _x=0.5,\delta _z=0.01$, $\beta =0.2$, $\epsilon =0.2$ and $\bar {x}_c =0$, $\bar {\psi }=45^\circ$. (a,d) Side views on vertical $x$–$z$ plane. The circle shows the trajectory of the centre of mass. The inset figures suggest the instantaneous, parabolic shape of the waterline near the floater. (b,c,e,f) Top views on $x$–$y$ plane. The arrows show the instantaneous force distribution along the floater that creates the instantaneous moment $K_z$. In (b,e), without considering the variable submersion depth, using $\bar {h}$. In (c,f), taking into account the variable submersion $\tilde {h}$.

We first consider the case where the depth variation is ignored (figure 12b,e), an approximation acceptable for small floaters only. At the wave crest the yaw angle is slightly larger than the mean value $\bar {\psi }$, whereas at the trough it is slightly smaller than $\bar {\psi }$. The lever arm, $\tilde {x} \sin \psi$, is therefore larger at the crest. We also see that the local force density $f_x$ is slightly larger at the crest than at the trough. Both effects impact the instantaneous moment in the same way. At the crest, the floater will experience a negative (clockwise) moment $K_z <0$ that is slightly larger than the positive (counterclockwise) moment $K_z >0$ in the troughs, explaining why the part $-\tilde {x} \sin \psi f_x \bar {h}$ in the integrand of $K_z$ is slowly pushing the floater towards the longitudinal position.

Figures 12(c,f) illustrates why including the varying submersion in the weighting of the force distribution changes this conclusion in the case of long floaters. We immediately see that there is a clear influence of the variable submersion at the tips of the floater: the weighted force density $f_x \tilde h$ is significantly changing in magnitude. As shown in figure 12(a,d), at the wave crest, the extremities of the floater are less submerged, whereas in the troughs they are more submerged. This locally changing submersion implies that the instantaneous moment is significantly decreased at the tips of the floater when they are at crest and increased at the tips when the floater is in a trough. The result for this floater with $F=125 > F_c$ is that the positive (counterclockwise) moment $K_z>0$ acting on the floater at the troughs is significantly larger than the negative moment at the crests, resulting in a slow rotation towards the transverse position.

In the introduction we mentioned that the yaw angle motion of small floaters with $F< F_c$ is analogous to that of the Kapitza pendulum, a pendulum with an oscillating anchor point (Kapitza Reference Kapitza1951; Landau & Lifschitz Reference Landau and Lifschitz1960; Butikov Reference Butikov2001). For such small floaters, we have seen that the variation of the immersion depth can be ignored and in that case (5.14) reduces to

(5.19)\begin{equation} F \ll F_c \ : \ \ddot{\psi} \approx{-} \epsilon s_\psi c_\psi \sin (\bar{x}_c-t) - \epsilon^2 s_\psi c_\psi \sin^2 (\bar{x}_c -t ). \end{equation}

Using the fact that $s_\psi c_\psi = (1/2) \sin 2 \psi$ and changing notation $2 \psi = \alpha$, we can rewrite this as

(5.20)\begin{equation} \ddot{\alpha} + \left( \underbrace{- \frac{\epsilon^2}{2}}_{\bar{g}} + \underbrace{\epsilon \sin (\bar{x}_c -t ) + O(\epsilon^2)}_{g'(t)} \right) \sin \alpha \approx 0 . \end{equation}

This is identical to the equation of motion for the angle $\alpha$ of a Kapitza pendulum, written in the frame of reference attached to the anchor point. The Kapitza pendulum, analogue to our floater, would be exposed to a weak $O(\epsilon ^2)$ downward external gravity $\bar {g}$ and a larger $O(\epsilon )$ oscillatory acceleration $g'(t)$ that is due to the motion of the anchor point. Using the same multiple time-scale techniques as previously, we can find that this pendulum has two equilibria. The lower position $\bar {\alpha } =0^\circ$ is stable and the top position $\bar {\alpha } = 180^\circ$ is unstable. Owing to the relation $\bar {\psi } = \bar {\alpha }/2$, this result is entirely equivalent to saying that short floaters prefer longitudinal positions, $\bar {\psi } = 0^\circ$, and avoid transverse positions, $\bar {\psi } = 90^\circ$.