2. Statement

Following Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007), we consider two-dimensional $O\bar{z}$ -symmetric steady-state surface waves excited by small-amplitude vertical harmonic motions of two rigidly connected rectangular hulls, ${\it\eta}_{3}(t)=-{\it\eta}_{3a}\cos ({\it\sigma}t)$ , in finite-depth water as shown in figure 1(a). The potential inviscid motions of incompressible liquid are assumed. The non-dimensional statement takes the gap width $L_{1}$ as the characteristic dimension and $1/{\it\sigma}$ as the characteristic time. This leads to the non-dimensional variables

(with the dimensional variables marked by an overbar), where $g$ is the gravity acceleration, ${\it\psi}(x,z,t)$ and $p(x,z,t)$ are the non-dimensional velocity potential and pressure, respectively, ${\it\rho}$ is the liquid density, $g$ is the gravitational acceleration and ${\it\epsilon}={\it\eta}_{3a}/L_{1}\ll 1$ . As a consequence, the distance between the rectangular bodies and the $Oz$ -axis becomes $1/2$ , the non-dimensional water depth is $h=\bar{h}/L_{1}$ , and the scaled dimensions of the stationary immersed rectangular body are $d=\bar{d}/L_{1}$ in the vertical (draft) direction and $B=\bar{B}/L_{1}=\bar{b}/L_{1}-1/2$ in the horizontal direction. The left normalized liquid domain is shown in figure 1(b).

Figure 2. Simplified schematic representations of: (a) small-amplitude oscillatory steady-state liquid motions through a slot of a slatted screen; (b) piston-mode sloshing in a rectangular moonpool according to the approximation of Molin (Reference Molin2001). The latter implies artificial sinks at $(\pm b_{\ast },-d)$ and an extended solid hull bottom for $b<|x|,\,z=-d$ . Fluid particles do not cross the lines $x=\pm b_{\ast },\,z<-d$ and $b_{\ast }<|x|<b,\,z=-d$ (excluding ends). Within the framework of our averaged pressure discharge approach, the vortex shedding from the sharp edges is assumed to be located mainly in zones $Z$ (i.e. two zones and one zone in cases (a) and (b), respectively). Dashed streamline components mark differences between (a) and (b) that disappear as the opening is approached, i.e. when $z\rightarrow -d,z<-d$ and $-b_{\ast }<x<b_{\ast }$ .

At a certain forcing frequency ${\it\sigma}_{\ast }$ (non-dimensional ${\it\Lambda}_{\ast }={\it\sigma}_{\ast }^{2}L_{1}/g$ by (2.1)), a resonant piston-mode sloshing occurs between the hulls. The inviscid potential flow statement well predicts ${\it\Lambda}_{\ast }$ , which can be computed using the method of Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007), for example, or approximated by the formula of Molin (Reference Molin2001). When deriving the latter formula, Molin artificially extended the rectangular hulls left and right to infinity but inserted two sinks at $\pm b_{\ast }$ (with $b_{\ast }>b$ ) on the extended hull bottom. The $b_{\ast }$ value is unknown a priori. When taking $b_{\ast }=b$ , the Molin formula gives a rough prediction of ${\it\Lambda}_{\ast }$ . However, if we already know ${\it\Lambda}_{\ast }$ (computed by another method), we can find

by inverting the formula (see equation (4.12) in Faltinsen et al. Reference Faltinsen, Rognebakke and Timokha2007). Molin’s solution will then give an approximation of the resonant piston-mode flows in the vicinity of the opening. Far from the opening, this approximation fails since it is assumed that there are no free-surface waves at infinity.

Figure 2 gives a schematic comparison of the local (at the opening) Molin’s piston-type sloshing (panel b) and an oscillatory steady-state flow through a slot in the slatted screen with a uniform (non-dimensional) harmonic approach velocity of $U(t)$ (panel a). The free surface in (b) remains far from the edges (provided by a finite draft $d$ ) so that the liquid flux per half-period is much less than the mean liquid volume inside the opening. Looking for a similarity in (a) implies the same for the liquid flux through a single slat, which is only possible for relatively small flux-velocity amplitudes and non-small slat thicknesses $2d$ . One can then introduce a fictitious almost-flat interface in the slot between the upper and lower liquid domains. The interface oscillates about the mean level $z=0$ and remains far away from the edges. The lower/upper liquid flow (separated by this interface) displays a similarity to the piston-type sloshing. Streamlines are shown by solid and dashed lines, with the solid lines demonstrating the flow similarity close to the openings. The flow separates at the inlet to the moonpool and at the outlet/inlet of the screen opening. We simplify the details of the vorticity field due to flow separation by locating it principally in zones $Z$ . Our primary objective is to express the consequence of flow separation in terms of a space-averaged (mean) pressure discharge across the moonpool without giving details on the vorticity field, which can in reality be rather complicated.

For the screen flow in figure 2(a), there is an analogy to $b_{\ast }$ at the middle of the slat, implying no cross-flows through the vertical lines $x=\pm b_{\ast }$ which confine the liquid flux through the chosen central slot. The solidity ratio is defined via $b_{\ast }$ as $Sn=(2b_{\ast }-1)/(2b_{\ast })$ . If $u_{0}(t)$ is the mean non-dimensional vertical (flux) velocity through the slot, i.e.  $u_{0}(t)=\int _{-1/2}^{1/2}v(x,t)\,\text{d}x$ , then the harmonic non-dimensional approach velocity can be expressed as $U(t)=u_{0}(t)/(2b_{\ast })$ and the Keulegan–Carpenter number as $K_{c}=2{\rm\pi}\bar{U}_{a}/(2{\it\sigma}\bar{b}_{\ast })={\rm\pi}u_{0a}/(2b_{\ast }^{2})$ , where the overbars imply dimensional values, $\bar{U}_{a}=\bar{u}_{0a}/(2\bar{b}_{\ast })$ (the dimensional amplitude of $\bar{U}(t)$ ) and $u_{0a}=2\bar{b}_{\ast }\bar{U}_{a}/(L_{1}^{2}{\it\sigma})$ is the non-dimensional amplitude. When following standard definitions, the spatial mean pressure discharge between the upper (–) and lower (+) liquid domains at each slot of the slatted screen is

(see equation (15) in Faltinsen et al. Reference Faltinsen, Firoozkoohi and Timokha2011) in the dimensional and non-dimensional (according to (2.1)) statements, respectively, where ${\it\rho}$ is the liquid density. The right-hand side of (2.3) shows that the pressure discharge depends on the mean flux velocity. Faltinsen et al. (Reference Faltinsen, Firoozkoohi and Timokha2011) demonstrated that, for sloshing problems, dealing with a complex non-uniform approach velocity field keeps the formulas applicable when these are based on the flux velocity  $u_{0}(t)$ .

Figure 2(b) schematically shows small-amplitude oscillatory flows associated with the piston-mode sloshing in a rectangular moonpool within the approximation framework of Molin (Reference Molin2001) and compares them with those in figure 2(a). In both cases, liquid particles have zero horizontal velocity component along the lines $x=\pm b_{\ast },\,y<-d$ and $b<|x|<b_{\ast },\,z=-d$ . In other words, the particles do not cross the vertical lines due to a symmetry (in the case of a) and because of the sinks at $x=\pm b_{\ast }$ (in the case of b), causing the particles to move along these lines. A difference is that liquid comes from infinity in case (a) but there are two sinks in case (b) and, as a consequence, the dashed streamline components do not coincide. However, the dashed streamline components disappear when approaching the structures (without ends), i.e. $z\rightarrow -d,\,z<-d$ and $-b_{\ast }<x<b_{\ast }$ ; therefore, one can say that the local velocity fields are similar at the openings (only solid streamline parts appear there). Moreover, the total liquid flux through the openings on $-1/2<x<1/2$ (or through the structure for $-b_{\ast }<x<b_{\ast }$ , without ends) is also determined by the same integral, $u_{0}(t)=\int _{-1/2}^{1/2}v(x,t)\,\text{d}x$ . Accounting for this local similarity (close to the openings) implies that the meanings of $Sn=(2b_{\ast }-1)/(2b_{\ast })$ and $K_{c}={\rm\pi}u_{0a}/(2b_{\ast }^{2})$ look similar in cases (a) and (b). Thus, if we know the function $K=(Sn,K_{c})$ for the screen case (a), we can use it to describe the pressure discharge in the moonpool opening IV – within the $1/2$ multiplier in (2.3b ) since case (b) is characterized by a single zone $Z$ : ${\rm\Delta}p(t)|_{moonpool}={\rm\Delta}p(t)/2$ .

The following equation and boundary conditions with respect to ${\it\psi}(x,z,t)$ remain unchanged and can be taken from Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007) (see the notation in figure 1 b):

where ${\it\Sigma}_{01}$ is the unperturbed free surface outside the moonpool opening, $S$ represents the mean wetted vertical walls of the stationary rectangle, $S_{B}$ is the horizontal seabed, $S_{D}$ is the bottom of the rectangular body, $S_{S}$ is the artificial vertical wall caused by the $Oz$ -symmetry and $n$ is the outer normal to the fluid boundary. The periodicity condition at infinity in (2.6) implies that we look for the steady-state periodic solution. The wavenumber $\mathscr{K}$ and function $F$ are unknown and should be found together with  ${\it\psi}$ .

Within the framework of our averaging pressure discharge technique, ${\rm\Delta}p(t)|_{moonpool}$ occurs at the keel level $T_{3}$ , i.e. between domains III and IV in figure 1(b). It leads to a time-dependent jump of the velocity potential which can be excluded (making ${\it\psi}$ continuous at $T_{3}$ ) via the standard substitution ${\it\psi}|_{IV}:={\it\psi}|_{IV}+C(t)$ with appropriate $C(t)$ . Derivations show that by proceeding in this way one changes the linear dynamic free-surface condition on ${\it\Sigma}_{02}$ :

is the kinematic condition (which remains unchanged) on ${\it\Sigma}_{02}$ , with $z=f(x,t)$ defining the free-surface elevation and with the $x$ -averaged flux velocity given by

Because we construct the theory to be applicable in a neighbourhood of ${\it\Lambda}_{\ast }$ , we use ${\it\Lambda}_{\ast }$ instead of ${\it\Lambda}$ in front of the nonlinear term in (2.7).

The difference between (2.4)–(2.8) and the standard free-surface problem (see Faltinsen et al. Reference Faltinsen, Rognebakke and Timokha2007) lies in the framed nonlinear integral term in the dynamic free-surface condition (2.7). A similar modified dynamic free-surface condition was used by Molin et al. (Reference Molin, Remy, Camhi and Ledoux2009) (‘fictitious plate model’) and Newman (Reference Newman2003) (‘lid methods’). Our statement suggests that we know a priori the non-dimensional resonant frequency ${\it\Lambda}_{\ast }$ , that $b_{\ast }$ is computed from (2.2) and that ${\it\Lambda}$ is close to ${\it\Lambda}_{\ast }$ . Obtaining ${\it\Lambda}_{\ast }$ implies finding ${\it\Lambda}$ for which the maximum wave response of (2.4)–(2.8) occurs without the framed term.

3. Numerical steady-state solution

Suggesting the equivalent linearization method, we will focus on finding the $\cos \,t$ and $\sin \,t$ harmonic components of the steady-state solution of (2.4)–(2.8). Supplementary materials (available at http://dx.doi.org/10.1017/jfm.2015.234) provide analytical and numerical details of the method of Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007) and present its modifications due to the extra nonlinear quantity in (2.7). We adopt the Baines–Peterson–Weisbach formula (Weisbach Reference Weisbach1855; Baines & Peterson Reference Baines and Peterson1951)

which expresses the pressure drop coefficient as a function of $Sn>0.3$ (controlled in numerical examples). The formula assumes that $K$ does not depend on the Keulegan–Carpenter number $K_{c}$ . Faltinsen et al. (Reference Faltinsen, Firoozkoohi and Timokha2011) successfully applied it to study sloshing in a screen-equipped tank. Improvement of (3.1) to include the $K_{c}$ effect was recently discussed by Hamelin et al. (Reference Hamelin, Love, Tait and Wilson2013).

Figure 3. Experimental case 1 of Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007) ( $B=1,d=1$ and $h=5.72222$ ): (a,c) theoretical and experimental maximum wave elevations of the piston-mode sloshing and the maximum wave elevations at $x=w11=3.88889$ , scaled by the forcing amplitude; (b,d) the corresponding phase shifts relative to the input sinusoidal motions of the hulls. In all panels the amplitude and phase results are plotted against ${\it\Lambda}$ , for a non-dimensional forcing amplitude of (a,b)  ${\it\epsilon}=0.013889$ ; (c,d)  ${\it\epsilon}=0.027778$ . The experimental data are taken from Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007): ▵, the measured maximum steady-state wave elevations inside the moonpool opening in (a,c) and the corresponding phase shifts in (b,d); ○, the measured maximum wave elevations and phase shifts at $x=3.88889$ ; ▿, the measured maximum elevations inside the opening filtered against the reflection effect of the outgoing wave (Kristiansen & Faltinsen Reference Kristiansen and Faltinsen2008). The dashed lines are taken from Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007) and represent the inviscid potential flow prediction. The solid lines represent our computations accounting for the $\sin \,t$ and $\cos \,t$ harmonics. Other numerical results are marked by filled triangles; they are taken from Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2008), who used a vortex-tracking model to account for the vortex-shedding effect at the lower moonpool edges.

Figure 4. Same as figure 3(a,c) except that the comparison is with experimental data reported by Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2012) (empty symbols). Following the original experimental publication, the amplitude and phase-shift results are plotted against the dimensional forcing period $T$ . The solid symbols show the numerical results of Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2012), obtained by applying a viscous solver to the liquid domain in the vicinity of the hulls: ▴, maximum wave elevation; ●, phase shift.

Figure 5. Same as figure 3 but for the experimental case 2 of Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007) (with $d=1.5$ ).

Comparisons of our new computations (solid lines) with experimental measurements (empty symbols) and nonlinear viscous simulations (filled symbols) are presented in figures 3–6. In addition, the dashed lines denote numerical results from Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007). We focus on the maximum wave amplitude inside the moonpool opening, the maximum steady-state wave elevations far from the hulls (at the control point $x=w11$ ) and the corresponding phase shifts (with respect to the $\sin t$ forcing). More details on the experimental cases, viscous solvers and notation used can be found in Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007), Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2008, Reference Kristiansen and Faltinsen2012) and Fredriksen et al. (Reference Fredriksen, Kristiansen and Faltinsen2015).

Figure 3 focuses on the experimental case 1 of Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007), where, under the same geometric input parameters, two different forcing amplitudes were tested. The caption explains the notation used. One should note that the measured maximum wave elevation inside the moonpool opening is, according to Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2008), affected by the wave reflection from a wall in the experimental wave basin. Other measurements, namely the maximum wave elevation at $w11$ and the phase shifts, were virtually uninfluenced by the reflection. Thus, along with symbols ▵ representing the actual measurements of maximum wave elevation in the moonpool opening, we have introduced inverted symbols ▿ to denote filtered output signals that are believed not to be affected by the reflection. Furthermore, Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2008) conducted nonlinear simulations that accounted for the vortex shedding at the sharp lower edges of the rigid hulls. Their numerical results are shown by filled triangles: the symbols ▾ represent numerical maximum wave elevations inside the moonpool opening, and the symbols ▴ mark numerical results for the phase shifts. What we see from the figure is that our new theoretical results are in satisfactory agreement with both the experiments and the computations of Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2008).

Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2012) reported additional (control) model tests with the same geometrical and physical input parameters as in case 1 of Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007). These model tests aimed at minimizing the aforementioned reflection effect. Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2012) also proposed a domain-decomposition method which suggests applying a nonlinear viscous solver to the fluid domain in the vicinity of the hull. We compare our theoretical output (solid lines) with the inviscid potential flow solution (dashed lines) and these additional experimental and viscous numerical results in figure 4. We see that our new approximation of the steady-state solution agrees satisfactorily with both experimental and numerical values.

To complete the validation procedure, we compare our new theoretical predictions with experimental cases 2 and 3 from Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007) in figures 5 and 6. Again, our approximation shows satisfactory agreement with the model tests and the nonlinear simulations by Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2008), especially for case 3.

Figure 6. Same as in figure 3 but for the experimental case 3 of Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2007) ( $B=1,d=0.5,h=2.86111,{\it\epsilon}=0.006944$ and $w11=1.94444$ ).