2 Nonlinear modal equations

3 Steady-state (periodic) solutions and their stability

The derived modal system facilitates analysing the steady-state wave regimes with a combined sway-surge-roll-pitch harmonic excitation. We start out with writing

Substituting (3.1) into (2.17) and (2.20) leads to the harmonic right-hand sides

in (2.17a ), (2.20a ) and (2.17b ), (2.20b ), respectively, where, accounting for (2.15),

The right-hand sides in the differential equations (2.17a ) and (2.17b ) governing the primary excited generalised coordinates $p_{11}(t)$ and $r_{11}(t)$ , respectively, can be rewritten in the form

where we denote, for brevity, $\unicode[STIX]{x1D716}_{x}=\unicode[STIX]{x1D716}_{pc}^{(1)},\bar{\unicode[STIX]{x1D716}}_{x}=\unicode[STIX]{x1D716}_{ps}^{(1)},\bar{\unicode[STIX]{x1D716}}_{y}=\unicode[STIX]{x1D716}_{rc}^{(1)}$ and $\unicode[STIX]{x1D716}_{y}=\unicode[STIX]{x1D716}_{rs}^{(1)}$ .

Because the periodicity condition allows for an arbitrary phase shift, $t:=t+t_{0}$ , using this shift makes it possible to find, without loss of generality, that

on the right-hand sides (3.4).

When looking for the periodic asymptotic solutions within (3.9), we follow the Bubnov–Galerkin procedure by Faltinsen & Timokha (Reference Faltinsen and Timokha2013) by posing the lowest-order asymptotic solution component

( $a,\bar{a},\bar{b}$ and $b$ are of $O(\unicode[STIX]{x1D716}^{1/3})$ ) of $p_{11}$ and $r_{11}$ as follows from the Narimanov–Moiseev asymptotics in § 2.2. The steady-state resonant sloshing can be classified by analysing the lowest-order-amplitude parameters $a,\bar{a},b$ and $\bar{b}$ . As we discussed above, the third-order generalised modes $p_{1n}(t)$ and $r_{1n},n\geqslant 2$ of (2.20) do not affect $p_{11}(t)$ and $r_{11}(t)$ and, therefore, only the inhomogeneous terms (3.4) influence the lowest-order-amplitude parameters. This means that, in view of the lowest-order component, the resonant sloshing regimes are functions of non-zero $\unicode[STIX]{x1D716}_{x},\bar{\unicode[STIX]{x1D716}}_{y}$ and $\unicode[STIX]{x1D716}_{y}$ .

One can introduce the artificial horizontal harmonic tank motions

which lead to the same (3.4), (3.5). These artificial motions define either longitudinal (when $\unicode[STIX]{x1D716}_{y}=0$ ) or elliptic (rotary) ( $\unicode[STIX]{x1D716}_{y}\neq 0$ ) harmonic tank forcing. The latter forcing is of the elliptic type since (3.7) implies that the tank horizontally moves along the elliptic trajectory given by

Suggesting, for clarity, that the semi-major axis of the elliptic forcing belongs to $Ox$ and assuming the elliptic motion occurs in the counter-clockwise direction, leads to

which can be assumed without loss of generality.

The second- and third-order generalised coordinates can be found from (2.18) and (2.19), (2.20), respectively. The second-order generalised coordinates take then the form

where

The third-order periodic generalised coordinates $p_{1n},n\geqslant 2$ and $p_{3n},n\geqslant 1$ are presented in appendix A by formulas (A 2) and (A 3), respectively.

Substituting (3.6) and (3.10) into equations (2.17) and gathering the first harmonic terms, $\cos \unicode[STIX]{x1D70E}t$ and $\sin \unicode[STIX]{x1D70E}t$ , gives the solvability (secular) equations

for $a,\bar{a},\bar{b}$ and $b$ , where

The secular system will further be solved in an analytical way. After finding $a,\bar{a},\bar{b}$ and $b$ from (3.12), the second- and third-order components of the asymptotic solution are fully determined by (3.10), (A 2), (A 3) and (A 1). Coefficients in this solution, as well as $m_{1}$ and $m_{3}$ in (3.12), are functions of $h$ , $r_{1}$ and the forcing frequency $\unicode[STIX]{x1D70E}$ . Utilising (2.15) shows that the latter dependence can be neglected by substituting $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{11}$ into the corresponding expressions. Dependence on $\unicode[STIX]{x1D70E}$ remains only in the $(\bar{\unicode[STIX]{x1D70E}}_{11}^{2}-1)$ quantity of (3.12).

One can follow Faltinsen & Timokha (Reference Faltinsen and Timokha2013) to study the stability of the asymptotic solution by using the linear stability analysis and the multi-timing technique. Limitations of the linear stability analysis scheme by the multi-timing technique are extensively discussed by Faltinsen & Timokha (Reference Faltinsen and Timokha2009, chaps. 8 and 9) and Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2003). We introduce the slowly varying time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D716}^{2/3}\unicode[STIX]{x1D70E}t/2$ (the order $\unicode[STIX]{x1D716}^{2/3}$ is chosen to match the lowest asymptotic terms in the multi-timing technique), Moiseev detuning (2.15) and express the infinitesimally perturbed solution

where $a,\bar{a},b$ and $\bar{b}$ are known and $\unicode[STIX]{x1D6FC},\bar{\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D6FD}$ and $\bar{\unicode[STIX]{x1D6FD}}$ are their relative perturbations depending on $\unicode[STIX]{x1D70F}$ . Inserting (3.14) into the Narimanov–Moiseev modal equations, gathering terms of the lowest asymptotic order and keeping linear terms in $\unicode[STIX]{x1D6FC},\bar{\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D6FD}$ and $\bar{\unicode[STIX]{x1D6FD}}$ lead to the following linear system of ordinary differential equations

where $\boldsymbol{c}=(\unicode[STIX]{x1D6FC},\bar{\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D6FD},\bar{\unicode[STIX]{x1D6FD}})^{\text{T}}$ and the matrix $\unicode[STIX]{x1D63E}$ has the elements

The instability of the asymptotic solution can be evaluated by studying (3.15). Its fundamental solution depends on the eigenvalue problem $\det [\unicode[STIX]{x1D706}\unicode[STIX]{x1D640}+C]=0$ , where $\unicode[STIX]{x1D640}$ is the identity matrix. Computations give the following characteristic polynomial

where $c_{0}$ is the determinant of $\unicode[STIX]{x1D63E}$ and $c_{1}$ is a complicated function of the elements of $\unicode[STIX]{x1D63E}$ . As discussed by Faltinsen & Timokha (Reference Faltinsen and Timokha2013), stability requires

When at least one of the inequalities is not fulfilled, the steady-state wave regime associated with the dominant amplitudes $a,\bar{a},b$ and $\bar{b}$ is not stable.

Miles (Reference Miles1984a ,Reference Miles b ) and Faltinsen et al. (Reference Faltinsen, Rognebakke and Timokha2003) classified the bifurcation points employing the same stability analysis scheme and using the information on the signs of $c_{i}$ . According to these works, $c_{0}$ vanishes at the turning point solutions and at the Poincaré bifurcation points, the zeros of the discriminant $c_{1}^{2}-4c_{0}$ are the Hopf bifurcation points where the real parts of a pair of complex-conjugate zeros of $c_{0}$ become positive. Definitions of the bifurcation points can also be found in Seydel (Reference Seydel2010).

Properties of the steady-state sloshing and its stability are determined by (3.12), which strongly depends on the forcing type (values of $\unicode[STIX]{x1D716}_{x}$ and $\unicode[STIX]{x1D716}_{y}$ ) as well as $m_{1}$ and $m_{3}$ . We assume the inequalities

which are numerically confirmed except for some isolated values of critical pairs $(h,r_{1})$ . The following equalities

can be considered as solvability conditions of (3.12). The conditions (3.20a ) and (3.20b ) show that $\bar{a}=\bar{b}$ and, if, in addition, $\unicode[STIX]{x1D716}_{y}\neq \unicode[STIX]{x1D716}_{x}$ (non-circular excitations), then $\bar{a}=\bar{b}=0$ and the only non-zero-amplitude parameters $a$ and $b$ are governed by

The solution of (3.21) depends on $m_{1}(h,r_{1})$ , $m_{3}(h,r_{1})$ and the forcing-amplitude parameters $\unicode[STIX]{x1D716}_{x}$ and $\unicode[STIX]{x1D716}_{y}$ by (3.4). As long as $h$ , $r_{1}$ , $\unicode[STIX]{x1D716}_{x}$ and $\unicode[STIX]{x1D716}_{y}$ are fixed and the pair $(h,r_{1})$ belongs to the zones I–VI in figure 6, the three-dimensional response curves $a=a(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}),b=b(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11})$ can be used for mathematical and/or physical treatment of the steady-state wave regimes. Several stable (and unstable) regimes may co-exist for a given forcing frequency. Their occurrence will in reality depend on initial and transient conditions.

The forthcoming numerical results are obtained with $I_{r}$ between 6 and 10 that always provides stabilisation to six to seven significant figures of the asymptotic steady-state solution as well as in $m_{1}$ and $m_{3}$ . Longitudinal, $\unicode[STIX]{x1D716}_{y}=0$ , elliptic, $0<\unicode[STIX]{x1D716}_{x}<\unicode[STIX]{x1D716}_{y}$ and rotary (circular), $0<\unicode[STIX]{x1D716}_{x}=\unicode[STIX]{x1D716}_{y}$ , forcing are analysed in §§ 4, 5 and 6, respectively.

4 Steady-state resonant sloshing with longitudinal forcing

The longitudinal harmonic forcing in the $Oxz$ plane means $\unicode[STIX]{x1D716}_{x}>0$ and $\unicode[STIX]{x1D716}_{y}=0$ in the secular system (3.21). The system has then only two analytical solutions well known from, for example, Gavrilyuk, Lukovsky & Timokha (Reference Gavrilyuk, Lukovsky and Timokha2000), Gavrilyuk et al. (Reference Gavrilyuk, Lukovsky, Trotsenko and Timokha2007). The first solution implies the so-called planar steady-state wave ( $\bar{a}=\bar{b}=b=0$ ). The non-zero lowest-order-amplitude parameter $a$ is governed by

This solution is characterised by the zero transverse wave component, namely, $r_{mi}(t)\equiv 0$ .

The second solution corresponds to swirling whose longitudinal ( $a\neq 0$ ) and transverse ( $b\neq 0$ ) amplitude parameters come from the system

Why this solution is called swirling is discussed by Faltinsen & Timokha (Reference Faltinsen and Timokha2009, chaps. 4 and 9). Ibrahim (Reference Ibrahim2005) names it the rotary wave but this definition is in our opinion not precise. Rotary (circular) motions normally suggest equal longitudinal and transverse wave components, $a=b$ in our asymptotic analysis, but (3.20c ) deduces $a\neq b$ for both longitudinal ( $\unicode[STIX]{x1D716}_{y}=0$ ) and elliptic ( $0<\unicode[STIX]{x1D716}_{y}<\unicode[STIX]{x1D716}_{x}$ ) excitations.

Let us first focus on the wave-amplitude response for an upright circular tank ( $r_{1}=0$ ) with a fairly deep liquid depth ( $1.5\lesssim h$ ) that can be associated with the experiments by Abramson et al. (Reference Abramson, Chu and Kana1966a ) ( $h=2$ , point C1 in figure 6) and Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) ( $h=1.5$ , C2). The latter experiments were done with diverse horizontally excited upright circular tanks and forcing frequencies close to the lowest natural sloshing frequency $\unicode[STIX]{x1D70E}_{11}$ . Abramson et al. (Reference Abramson, Chu and Kana1966a ) and Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) document a series of observations pointing out planar waves, swirling and irregular wave patters in certain frequency ranges. Measurements of the steady-state wave elevations near the walls, as well as of the horizontal hydrodynamic force (Abramson et al. Reference Abramson, Chu and Kana1966a ), were reported. All the model tanks had radii $\bar{r}_{2}$ between 8 and 18 cm. This means that the Bond number $Bo$ is approximately 1000. As we mentioned in § 2.4, the Bond number should be of the order $10^{4}\lesssim Bo$ to fully neglect the surface tension in the multimodal method, except at the wall where the dynamic contact line (meniscus) effect, run-up and overturning can matter (see photos in Abramson Reference Abramson1966; Royon-Lebeaud et al. Reference Royon-Lebeaud, Hopfinger and Cartellier2007). The case $Bo\approx 1000$ implies that the surface tension may also quantitatively affect the measured wave elevations slightly away from the wall but one can neglect it for the hydrodynamic forces (see p. 125 in Faltinsen & Timokha Reference Faltinsen and Timokha2009). The experimental observations and measurements of Abramson et al. (Reference Abramson, Chu and Kana1966a ) and Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) will be used for validating the derived modal system and the corresponding asymptotic steady-state results.

Figure 7.

Typical response curves (in terms of the lowest-order-amplitude parameters $|a|$ and $|b|$ ) for an upright circular tank with longitudinal harmonic forcing in the $Oxz$ -plane and a fairly deep depth, $1.5\lesssim h$ . The branching is computed by using (4.1) for planar and (4.2) ( $\unicode[STIX]{x1D716}_{x}=P_{1}\unicode[STIX]{x1D705}_{11}\unicode[STIX]{x1D702}_{1a}$ ) for swirling steady-state waves; $h=2,r_{1}=0$ and the horizontal forcing amplitude $\unicode[STIX]{x1D702}_{1a}=0.02~(\unicode[STIX]{x1D707}_{5a}=0)$ (associated with experimental input by Abramson et al. Reference Abramson, Chu and Kana1966a ). The three-dimensional branching in the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|a|,|b|)$ -space is shown in panel (c). Its projections on the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|a|)$ and $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|b|)$ planes are presented in (a) and (b), respectively. The three-dimensional curves ( $|b|>0$ ) correspond to swirling but the planar ones (belonging to the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|a|)$ -plane) imply planar steady-state waves. The solid lines mark the stable steady-state waves. The panel (a) specifies frequency ranges of stable planar and swirling waves. There are no stable steady-state waves between ordinates of $U$ (turning point) and $H$ (the Hopf bifurcation point corresponds to the zero of $c_{1}^{2}-4c_{0}$ in (3.18) so that the real parts of a pair of complex-conjugate zeros of $c_{0}$ become positive to the left of $H$ ) where an irregular sloshing is expected. To the right of $P$ (the Poincaré bifurcation point emanates two unstable branches for swirling and one stable branch, which implies the stable planar wave), stable swirling and planar waves co-exist but, to the left of $U$ , only planar waves are stable.

Typical theoretical three-dimensional response curves associated with experiments by Abramson et al. (Reference Abramson, Chu and Kana1966a ) ( $h=2,r_{1}=0$ with $\unicode[STIX]{x1D702}_{1a}=0.02,\unicode[STIX]{x1D707}_{5a}=0$ ) are drawn in figure 7 in terms of the lowest-order-amplitude parameters $a$ and $b$ . The amplitude parameters are computed by using (4.1) and (4.2). The stable periodic (steady-state) solutions are denoted by the solid lines and the dashed lines are used to mark the unstable ones. There are three panels in the figure. The panel (c) demonstrates the three-dimensional response curves in the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|a|,|b|)$ -space but other panels (a and b) show projections of the branching on the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|a|)$ and $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|b|)$ planes. The planar steady-state waves of (4.1) are easily distinguished in (c) as belonging to the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|a|)$ -plane. The three-dimensional response curves ( $b\neq 0$ ) correspond to swirling. The branching contains three bifurcation points $U$ , $H$ and $P$ whose positions determine the effective frequency ranges where stable planar, swirling or irregular waves are theoretically expected. This fact is illustrated in (a). The forcing frequencies to the left of $U$ lead to planar steady-state waves. In the frequency range between $U$ and $H$ , both planar and swirling waves are unstable and one should expect irregular, chaotic wave patterns where switches between planar and swirling occur on a long time scale (the range is marked as ‘irregular’). In the frequency range between $H$ and $P$ , only stable swirling exists, but the forcing frequencies on the right of $P$ may lead to either planar or swirling steady-state waves depending on the initial transients.

Figure 8.

Theoretical (solid lines) and experimental (circles) estimates of bounds between the frequency ranges where planar, irregular and swirling waves occur. The bounds are associated with ordinates of the three bifurcation points in figure 7(a): $U$ (the planar wave becomes unstable with increasing of either $\unicode[STIX]{x1D70E}$ or $\unicode[STIX]{x1D702}_{1a}$ , the solid circles), $P$ (the planar wave becomes unstable with decreasing of $\unicode[STIX]{x1D70E}$ and/or increasing $\unicode[STIX]{x1D702}_{1a}$ , the solid circles) and $H$ (swirling becomes unstable with decreasing $\unicode[STIX]{x1D70E}$ , the empty circles). The frequency ranges of irregular waves and swirling increase with increasing forcing amplitude $\unicode[STIX]{x1D702}_{1a}$ . The experiments by Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) are shown in (a) and (b) corresponds to Abramson et al. (Reference Abramson, Chu and Kana1966a ).

Positions of $U$ , $P$ and $H$ weakly vary with the liquid depth $h$ as $1.5\lesssim h$ . However, they strongly depend on the forcing amplitude. The latter dependence was studied in experiments by Abramson et al. (Reference Abramson, Chu and Kana1966a ), Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) and Hopfinger & Baumbach (Reference Hopfinger and Baumbach2009) who estimated bounds between planar waves, swirling and irregular waves in the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},\unicode[STIX]{x1D702}_{1a})$ -plane. We compare our theoretical predictions with these experimental bounds in figure 8. The panel (a) is devoted to the case of Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) (point C2 in figure 6) and the panel (b) corresponds to Abramson et al. (Reference Abramson, Chu and Kana1966a ) (point C1). The theoretical bounds (positions of the bifurcation points) are marked by the solid lines. The solid circles give experimental bounds for the planar waves (associated with positions of $U$ and $P$ ) but empty circles evaluate the experimental border between swirling and chaotic sloshing (point $H$ in figure 7).

Figure 9.

Theoretical and measured steady-state non-dimensional maximum horizontal hydrodynamic force (along the excitation direction) and ‘average’ amplitude of elevations at the wall versus the forcing frequency for the planar wave regime. The measurements are taken from Abramson et al. (Reference Abramson, Chu and Kana1966a ). The theoretical curves are marked by the non-dimensional horizontal forcing amplitudes $\unicode[STIX]{x1D702}_{1a}$ . The experimental symbols are: ♢ –  $\unicode[STIX]{x1D702}_{1a}=0.0112$ , ▿ – 0.023, ▫ – 0.0344 and ▵ – 0.0454. The non-dimensional horizontal hydrodynamic force amplitude is computed as $\max _{0\leqslant t<2\unicode[STIX]{x03C0}}100|F_{1}(t)|/(\unicode[STIX]{x1D70C}\bar{g}(2\bar{r}_{2})^{3})$ by (2.21). According to Abramson et al. (Reference Abramson, Chu and Kana1966a ), ‘average amplitude at the wall’ means one-half of the wave height near the wall. The distance from the wall was not specified, and the theoretical values are computed at $(x,y)=(1,0)$ .

Figure 10.

Theoretical and measured non-dimensional maximum steady-state wave elevations (the maximum is also taken from values at the two probes, which are located at $(x,y)=(0.875,0)$ and $=(0,0.875)$ ) versus the forcing frequency $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}$ and the horizontal forcing amplitudes $\unicode[STIX]{x1D702}_{1a}$ . The results are associated with two experimental series by Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007, experimental data from the original figures 2 (now, a) and 7 (now, b)). Empty symbols correspond to measurements done for the experimental planar regime but the filled symbols imply swirling. The theoretical curves are marked by $\unicode[STIX]{x1D702}_{1a}$ which are 0.0033 (♢, ♦), 0.0066 (▿, ▾), 0.0133 (▫, ▪), 0.023 (▹, ▸), 0.0266 (▵, ▴) and 0.045 (◃, ◂).

Figure 8(a) shows that our theoretical prediction of the effective frequency ranges is supported by the experiments of Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007). The theoretical results are also consistent with the experimental data by Abramson et al. (Reference Abramson, Chu and Kana1966a ) for the stability bound, which is associated with $P$ ( $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}>1$ ). This is seen in panel (b). Here, we see that experimental symbols for $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}<1$ (the bound is associated with the turning point $U$ ) are clearly inconsistent with our theory and with the experiments by Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007). We believe that this inconsistency can be clarified by different experimental techniques in Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) and Abramson et al. (Reference Abramson, Chu and Kana1966a ). Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) detected the bounds (positions of $P$ , $U$ and $H$ ) by conducting the model tests with a fixed forcing amplitude $\unicode[STIX]{x1D702}_{1a}$ and a stepwise change of the forcing frequency $\unicode[STIX]{x1D70E}$ to reach the steady-state condition for each step. Dealing with the planar wave regime, positive steps of $\unicode[STIX]{x1D70E}$ were used to detect $U$ and negative ones for $P$ . As a matter of fact, Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) path followed the stable branches that are described above. Abramsons’ experiments on detecting the bounds of the stable planar waves were performed with a fixed $\unicode[STIX]{x1D70E}$ and a stepwise increase of $\unicode[STIX]{x1D702}_{1a}$ . Appearance of swirling-type wave patterns was used as a signal that the planar wave is unstable for the last tested $\unicode[STIX]{x1D702}_{1a}$ . The experimental steps of $\unicode[STIX]{x1D702}_{1a}$ were not documented. Because the response curve at the turning point $U$ is very steep (see figure 7), we believe the experimental strategy of Abramson et al. (Reference Abramson, Chu and Kana1966a ) requires that these steps should be small enough to avoid three-dimensional transient waves, which could be wrongly interpreted as passage to steady-state swirling.

Whereas using the five-dimensional modal Lukovsky’s modal system can give theoretical results that are very close to those in figure 8 (see figure 9.29 in Faltinsen & Timokha Reference Faltinsen and Timokha2009, the difference is almost invisible if superposing the curves), the forthcoming results on the hydrodynamic force and, especially, the wave elevation near the wall are quite sensitive to the number of the second- and third-order modes (determined by the integer $I_{r}$ ). Normally, we have to choose $I_{r}$ from 6 to 10 to stabilise to six to seven significant figures in the coefficients of the periodic solution from appendix A, which was a theoretical base for computing the force and elevation. This sensitivity to $I_{r}$ with $I_{\unicode[STIX]{x1D703}}=3$ explains why Lukovsky (Reference Lukovsky2015, figures 8.3(b) and 8.11) did not compare (by referring to a possible measurement error) his steady-state result by the five-dimensional model ( $I_{\unicode[STIX]{x1D703}}=2,I_{r}=1$ ) with several measurements of Abramson et al. (Reference Abramson, Chu and Kana1966a ). The difference was significant.

Figure 9 compares the theoretical steady-state non-dimensional horizontal hydrodynamic force $F_{1}(t)$ by (2.21) and the theoretical average amplitude defined as one-half of the wave height of the non-dimensional wave elevations near the wall. The theoretical curves are marked by the $\unicode[STIX]{x1D702}_{1a}$ -values. The theoretical steady-state result for the hydrodynamic force is supported by these experiments. We see it in (a). A discrepancy is only seen for $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}>1.1$ with the larger experimental forcing amplitudes $\unicode[STIX]{x1D702}_{1a}=0.0454$ and 0.0344. Abramson et al. (Reference Abramson, Chu and Kana1966a ) did not specify the distance of the measurement probe from the wall mentioning, however, that the probe was not located exactly at the wall. Since the distance is unknown, we computed the average amplitude exactly at the wall, at $(x,y)=(1,0)$ . The computed values are expected to be larger than at the actual experimental distance from the wall. This partly explains why the theoretical values in panel (b) are larger than the measurements. The results are in satisfactory agreement, except for the smallest amplitude, $\unicode[STIX]{x1D702}_{1a}=0.0112$ and $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}<0.83$ , when the hydrodynamic force is well predicted in (a) but the measured average amplitude is approximately two times lower than the theoretical values in (b). Specifically, the linear sloshing results almost coincide (the difference is less than 1 %) with our steady-state planar wave prediction for $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}\leqslant 0.8$ and $1.2\leqslant \unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}$ .

Figure 11.

The response curves (similar to figure 7) for the annular upright tank with $r_{1}=0.25$ , $h=2$ and $\unicode[STIX]{x1D702}_{1a}=0.02$ . Extra bifurcation points $H_{i}$ appear instead of the single $H$ . Condition (4.3) is not satisfied but $O(1)=m_{1}>0$ and $O(1)=m_{3}>0$ . The hard-spring-type response curves are detected for both planar and swirling steady-state waves. The planar sloshing is stable for any $\unicode[STIX]{x1D70E}$ in a neighbourhood of the primary resonance and there is no ‘irregular’ wave.

Figure 12.

The response curves (similar to figure 7) for the annular upright tank with $r_{1}=0.35$ , $h=2$ and $\unicode[STIX]{x1D702}_{1a}=0.02$ . Extra bifurcation points $U_{1}$ and $H_{1},H_{2}$ appear instead of $H$ . Condition (4.3) is not satisfied but $O(1)=m_{1}<0$ and $O(1)=m_{3}<0$ . The soft-spring-type response curves are detected for both planar and swirling steady-state waves. The planar sloshing is stable for any $\unicode[STIX]{x1D70E}$ in a neighbourhood of the primary resonance and there is no ‘irregular’ wave.

Abramson et al. (Reference Abramson, Chu and Kana1966a ) do not present wave measurements for swirling but Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) document the maximum free-surface elevation in their two independent experimental series (see figures 2 and 7 of the original paper by Royon-Lebeaud et al. Reference Royon-Lebeaud, Hopfinger and Cartellier2007). We compare our theory and the experimental values in figure 10. Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) indicate positions of the two measurement probes ( $r=0.875\bar{r}_{2}$ , $\unicode[STIX]{x1D703}=0$ and $\unicode[STIX]{x03C0}/2$ ) which are located at the $1/8\bar{r}_{2}$ distance from the wall to avoid the effect of the surface tension, overturning and other near-wall phenomena. Having known, qualitatively, the response curves branching (they refer to Faltinsen et al. Reference Faltinsen, Rognebakke and Timokha2003), Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007) path followed the theoretical expectations with a stepwise change of the forcing frequency while the forcing amplitude was fixed. This experimental approach made it possible to detect both planar and swirling steady-state waves in the frequency range where these co-exist. Figure 10 shows that our steady-state asymptotic theoretical results are in general supported by the experiments. The maximum relative difference is found for the lower experimental forcing amplitude and the difference increases away from the primary resonance $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}=1$ . This is a quite curious fact since the nonlinearity and the free-surface phenomena should be less important for smaller forcing amplitudes and away from the primary resonance zone. A logical explanation is that the actual dimensional values of these elevation amplitudes are approximately 1 mm and may be difficult to measure.

Figure 10 focuses on the steady-state wave amplitudes for swirling. After reading Royon-Lebeaud et al. (Reference Royon-Lebeaud, Hopfinger and Cartellier2007, § 4.3) who reported various viscous-type free-surface phenomena including breaking and overturning waves, we expected to see only qualitative agreement between our theory and the experimental data. However, the figure shows a satisfactory quantitative agreement. Breaking waves and transients in the higher modes (see a discussion on the higher modes affect for a square base tank by Faltinsen et al. Reference Faltinsen, Rognebakke and Timokha2006) should lead to a larger wave elevation near the wall than in the steady-state condition. We see that in the figure. However, the relative difference does not look dramatic. It is rather of a stochastic character that is typical when experimental transients do not completely die out.

Experimental studies on the steady-state sloshing in an annular upright tank are rare. The authors only recall the recent paper by Takahara & Kimura (Reference Takahara and Kimura2012). However, these experiments, as explained in § 2.4, were done for $(h,r_{1})$ outside zones I–VI in figure 6. This means that our modal theory cannot be applied to model the experiments by Takahara & Kimura (Reference Takahara and Kimura2012). An adaptive modal theory is required.

The response curves presented in figure 7 are typical for the longitudinal forcing and annular upright tanks when the coefficients $m_{1}(h,r_{1})$ and $m_{3}(h,r_{1})$ in the secular system (3.12) satisfy the conditions

This condition holds true for the circular cross-section ( $r_{1}=0$ ) and $1.5\lesssim h$ . Numerous calculations were done for $1.5\leqslant h\leqslant 2$ and $r_{1}\neq 0$ throughout the zones I–VI in figure 6 to justify that (4.3) is fulfilled in I, II, V and VI but, generally speaking, it may not be satisfied in III and IV. An alternative branching is possible in the latter zones. This fact is exemplified in figures 11 and 12 for $r_{1}=0.25$ (point E1 in figure 6) and $r_{1}=0.35$ (E2), respectively. In these examples, the planar steady-state wave is stable for any resonant forcing frequency $\unicode[STIX]{x1D70E}$ and, therefore, irregular waves do not occur. Stable swirling co-exists with stable planar waves always slightly away from the primary resonance $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}=1$ . This means that using an annular tank with these radii ( $r_{1}\approx 0.25$ or 0.35) may help to avoid both irregular and swirling waves.

5 Steady-state resonant sloshing with purely elliptic forcing

In that case, $0<\unicode[STIX]{x1D716}_{y}<\unicode[STIX]{x1D716}_{x}$ equality (3.20c ) shows that the conditions $ab\neq 0$ and $a\neq b$ should be satisfied. The non-zero amplitude parameters $a$ and $b$ are governed by (3.21) which determines swirling for all input parameters. The authors do not know how to find an analytical solution of (3.21). A numerical procedure is applied after rewriting (3.21) to the equivalent form

The first equality is a depressed cubic with respect to $b$ whose coefficient at the linear term is a function of $a$ . The second equality computes the forcing frequency, $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}$ ( $\bar{\unicode[STIX]{x1D70E}}_{11}^{2}-1$ ), as a function of $a$ and $b$ . The numerical procedure suggests varying $a$ in an admissible range, solving the depressed cubic (finding $b=b(a)$ ) and computing $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}$ as a function $a$ and $b=b(a)$ . When solving the depressed cubic, one should check for the discriminant

Cartano’s theorem states that, (i) if $\unicode[STIX]{x1D6E5}>0$ , then there are three distinct real roots for $b$ , (ii) if $\unicode[STIX]{x1D6E5}=0$ , then the equation has at least one multiple root and all its roots are real and (iii) if $\unicode[STIX]{x1D6E5}<0$ , then the equation has one real root and two non-real complex conjugate roots.

When considering $\unicode[STIX]{x1D6E5}$ as a function of $a$ , a simple analysis shows that, if $m_{1}-m_{3}<0$ , there exists only a negative real root $a_{\ast }<0$ of $\unicode[STIX]{x0394}(a_{\ast })=0$ so that $\unicode[STIX]{x0394}(a)>0$ for $a<a_{\ast }$ and $0<a$ (three real solutions) but $\unicode[STIX]{x0394}(a)<0$ for $a_{\ast }<a<0$ (a unique real solution). Analogously, if $m_{1}-m_{3}>0$ , there exists only a positive real root $a_{\ast }>0$ of $\unicode[STIX]{x0394}(a_{\ast })=0$ so that $\unicode[STIX]{x0394}(a)>0$ for $a<0$ and $a_{\ast }<a$ but $\unicode[STIX]{x0394}(a)<0$ for $0<a<a_{\ast }$ .

Figure 13.

Response curves from figure 7 are re-drawn accounting for the signs of $a$ and $b$ . The solid lines denote stable sloshing and the dotted lines unstable ones. The three-dimensional view in the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a,b)$ (panel d) space and their projections on the planes $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a)$ , $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},b)$ (panels a and b) and $(a,b)$ (panel c). The branches are denoted by $b_{i},i=1,\ldots ,5$ . The bifurcation point $P$ connects the branches $b_{2}$ and $b_{5}$ .

Figure 14.

The response curves for the elliptic forcing with $\unicode[STIX]{x1D716}_{y}=0.2\cdot \unicode[STIX]{x1D716}_{x}$ when other input parameters are taken from figure 13. The bifurcation point $P$ disappears after splitting $b_{2}$ and $b_{5}$ into two non-connected branches.

Figure 15.

The same as in figure 14 but for $\unicode[STIX]{x1D716}_{y}=0.8\cdot \unicode[STIX]{x1D716}_{x}$ .

Results on the longitudinal forcing from § 4 ( $\unicode[STIX]{x1D716}_{y}=0$ ) can be computed by using the aforementioned numerical scheme. The limit $\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}\rightarrow 0$ continuously transforms the system (5.1) to (4.1) and (4.2). The depressed cubic equation from (5.1) has then the zero solution ( $b=0$ , planar waves) and two non-zero solutions with opposite signs ( $\pm b\neq 0$ , swirling). The latter two ( $\pm b$ ) solutions correspond to two identical steady-state swirling waves propagating in two opposite angular directions. When analysing the longitudinally excited waves, the signs of $b$ (and $a$ ) were not important. That is why the response curves were drawn in the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},|a|,|b|)$ -space. For the elliptic forcing, we do not have those two identical (to within the angular direction, $\pm b$ ) swirling waves. The signs of $b$ are important, even though the minor-axis forcing component is relatively small. Our positive $\unicode[STIX]{x1D716}_{x}$ and $\unicode[STIX]{x1D716}_{y}$ implies that the elliptic forcing occurs in the counter-clockwise direction (in the top view). When the lowest-order-amplitude parameters satisfy the inequality $ab>0$ , the corresponding swirling wave is co-directed with the forcing direction but the inequality $ab<0$ implies that the swirling wave is counter-directed to the forcing.

In the forthcoming numerical examples, we adopt the input data, $m_{1},m_{3}$ and $\unicode[STIX]{x1D716}_{x}$ from figure 7 corresponding to an upright circular tank with $h=2$ and the major-axis forcing component is $\unicode[STIX]{x1D702}_{1a}=0.02$ . The minor-axis component $\unicode[STIX]{x1D702}_{2a}$ will change from zero to 0.02. The theoretical results should remain qualitatively the same for any other admissible $r_{1}\neq 0$ and $1.5\lesssim h$ from zones I, II, V, VI in figure 6 where the condition (4.3) holds true.

The response curves branching changes with $0\leqslant \unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}<1$ . Our starting point is the branching from figure 7 (the longitudinal forcing, $\unicode[STIX]{x1D716}_{y}=0$ ) which is redrawn in figure 13 by accounting for the signs of $a$ and $b$ . The figure shows the three-dimensional view in the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a,b)$ space as well as projections on the planes $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a)$ , $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},b)$ and $(a,b)$ . As earlier, the solid lines denote the stable steady-state solution and the dotted lines correspond to the unstable ones. The branches are marked by $b_{i},i=1,\ldots ,5$ . Notations for the bifurcation points are the same as in figure 7. The branches $b_{2}$ and $b_{5}$ are connected at the bifurcation point $P$ . All the branches are symmetrically located with respect to the $b=0$ plane. The Hopf bifurcation point $H$ in figure 7 splits, formally, into $H_{+}$ and $H_{-}$ (because of the two opposite swirling angular directions). The branches $b_{3}$ and $b_{4}$ coincide with each other in the projections on the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a)$ plane.

Passage to the elliptic forcing with a non-zero $\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}$ breaks down the $\pm b$ -symmetry of the branching. We illustrate this fact in figure 14 drawn with the same input data but for $\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}=0.2$ . Because $ab\neq 0$ , no planar waves exist along all the branches. The Poincaré bifurcation point $P$ disappears and branches $b_{2}$ and $b_{5}$ are not connected. The frequency range of irregular waves still exists but becomes narrower. Stable counter-directed swirling ( $ab<0$ ) is possible. The corresponding solutions are on the solid lines of $b_{4}$ . All other solid lines imply swirling, which is co-directed with the forcing direction.

The branching from figure 14 remains qualitatively the same with increasing $\unicode[STIX]{x1D716}_{y}$ . This is illustrated in figure 15 where $\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}=0.8$ . However, there is an important difference – which is vanishing of the frequency range where irregular waves are expected. This frequency range is represented in this figure by a narrow interval between the ordinates $H_{+}^{(1)}$ and $H_{+}^{(2)}$ . Another tendency is that the stable periodic solutions on the branches $b_{1},b_{2}$ and $b_{3}$ correspond to a nearly rotary (a co-directed circular swirling) wave which is characterised by almost-equal amplitude parameters $a\approx b$ (we recall that $a$ and $b$ characterise the two lowest-order perpendicular wave components). This fact is especially transparently seen in the projections on the $(a,b)$ -plane. On the other hand, stable counter-directed (to the forcing direction) swirling waves still exists in a wide frequency range. This range is associated with the solid lines on $b_{4}$ . Will the range disappear/vanish with in the limit $1>\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}\rightarrow 1$ ? We have conducted extensive numerical experiments testing the minor-axis components up to $\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}=0.9999$ . These computations showed that the frequency range of the counter-directed swirling remains almost the same and the range always includes the primary resonant point $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{11}$ . However, the next section will show that there are no stable counter-directed swirling waves for purely rotary forcing when the exact equality $\unicode[STIX]{x1D716}_{y}=\unicode[STIX]{x1D716}_{x}$ is fulfilled. This means that the response curves branching is not a continuous function of $\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}$ at 1. The latter fact will be discussed in the next section.

6 Steady-state resonant sloshing for rotary forcing

When $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D716}_{x}=\unicode[STIX]{x1D716}_{y}$ , equations (3.20a ) and (3.20b ) are unable to derive that $\bar{a}$ and $\bar{b}$ are zeros but deduce, instead, the equality $\bar{a}=\bar{b}=c$ . The latter makes $\unicode[STIX]{x2462}\equiv \unicode[STIX]{x2463}$ in (3.12). By taking the sum $\unicode[STIX]{x2460}+\unicode[STIX]{x2461}$ and the difference $\unicode[STIX]{x2460}-\unicode[STIX]{x2461}$ , we transform (3.12) to the form

in which the two homogeneous equations contain identical expressions in the square bracket. These expressions are multiplied by $(a-b)$ and $c$ , respectively.

In the forthcoming analysis, we adopt $a_{+}=(a+b)/2,a_{-}=(a-b)/2$ instead of $a$ and $b$ . When both $a_{-}$ and $c$ are zeros, we arrive at

which imply rotary (circular swirling) waves characterised by equal longitudinal (along $Ox$ ) and transverse (along $Oy$ ) amplitude components, $p_{11}(t)=a_{+}\cos (\unicode[STIX]{x1D70E}t)+O(\unicode[STIX]{x1D716}),r_{11}(t)=a_{+}\sin (\unicode[STIX]{x1D70E}t)+O(\unicode[STIX]{x1D716})$ . The rotary waves are co-directed with the circular (rotary) forcing. The stability analysis with $m_{1}$ and $m_{3}$ satisfying (4.3) shows that the rotary wave by (6.2) is stable for all the resonant forcing frequencies.

When either $a_{-}\neq 0$ or $c\neq 0$ , the two identical square bracket expressions of (6.1) must be zero. This makes the second and third equalities of (6.1) automatically satisfied and, therefore, three amplitude parameters $a_{+},a_{-}$ and $c$ should be found from the two equalities

which define the following lowest-order steady-state solution

The amplitude parameter $a_{+}$ can be found from the first equation of (6.3) but the amplitude parameters $a_{-}$ and $c$ are not uniquely defined. Only the sum $a_{-}^{2}+c^{2}$ can be found for any fixed pair $(\bar{\unicode[STIX]{x1D70E}}_{11}^{2},a_{+})$ from the first cubic equation. This defines a manifold $a_{+}=a_{+}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11}),a_{-}^{2}+c^{2}=F(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a_{+})$ in the four-dimensional space $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a_{+},a_{-},c)$ . Numerical analysis of the solution (6.4) shows that it is unstable on the aforementioned manifold due to $c_{0}=0$ in the characteristic equation (3.17).

Figure 16.

The response curves branching in projections on the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a_{+})$ and $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a_{-})$ planes for the elliptic forcing with $\unicode[STIX]{x1D716}_{y}=0.95\unicode[STIX]{x1D716}_{x}$ (panels a and b) and the rotary forcing $\unicode[STIX]{x1D716}_{y}=\unicode[STIX]{x1D716}_{x}$ (panels c and d). Other input parameters are the same as in figure 14.

When $c=0$ , system (6.3) defines the three-dimensional response curves $a_{+}=a_{+}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11})$ , $a_{-}=a_{-}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11})$ which imply the solution

which has the same form as for the elliptically excited swirling with $\unicode[STIX]{x1D716}_{y}<\unicode[STIX]{x1D716}_{x}$ . Figure 16 compares the response curves branching for the elliptic, almost-rotary forcing ( $\unicode[STIX]{x1D716}_{y}/\unicode[STIX]{x1D716}_{x}=0.95$ in the calculations, a) and the branching for the rotary forcing $\unicode[STIX]{x1D716}_{y}=\unicode[STIX]{x1D716}_{x}$ (b). The result is presented in terms of the amplitude parameters $a_{+}$ and $a_{-}$ and in projections on the $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a_{+})$ and $(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}_{11},a_{-})$ planes. Coefficients $m_{1}$ and $m_{3}$ are taken from the case of an upright circular tank with $h=2$ and $\unicode[STIX]{x1D702}_{1a}=0.02$ (the branchings remain qualitatively the same for other forcing amplitudes and $m_{1},m_{3}$ satisfying (4.3)). As we remarked above, the stable solutions on $b_{4}$ (the counter-directed swirling) exist for the elliptic forcing but does not exist in the limit case. Another important conclusion is that two different branches $b_{1}$ and $b_{3}$ (a) become connected at the bifurcation point $P$ (in b) so that stable sub-branches of $b_{1}$ and $b_{3}$ imply the rotary (circular) wave with $a_{-}=0$ ( $b_{1}+b_{3}$ in b) but unstable points on these branches constitute the branch $\bar{b}_{1}+\bar{b}_{3}$ crossing the stable sub-branch.