# Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion

## Abstract

The coupled motion—between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics—is considered. The fluid layers are assumed to be thin and the shallow-water assumption is applied. The governing form of the Lagrangian functional in the Lagrangian particle path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two- and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two- and three-layer cases, based upon the *implicit-midpoint rule*. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani et al. (J Fluid Struct 59:432–460, 2015) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.

## Keywords

Dynamic Lagrangian particle path Multilayer Sloshing Two-layer## 1 Introduction

*h*(

*x*,

*t*) is the fluid depth,

*u*(

*x*,

*t*) is the depth averaged horizontal velocity component, \(g>0\) is the gravitational constant and the subscripts denote partial derivatives. Transforming to the LPP formulation gives

*x*(

*a*,

*t*). Moreover the Eq. (1.3) is the Euler–Lagrange equation deduced, with fixed endpoint variations, from the Lagrangian functional

The aim of this paper is to derive the LPP formulation to shallow water flows, with multiple layers of differing density, in a vessel with dynamic coupling, and use it as a basis for a variational formulation and numerical scheme. Although this generalisation is straightforward in principle, the variational formulation has complex subtleties due to the integration over different label spaces. Stewart and Dellar [3] successfully developed a variational formulation for shallow-water multilayer hydrodynamics by starting with a variational formulation for the full three-dimensional problem and reducing. The resulting variational principle for shallow water involves integration over each layer with respect to the local labels. With an aim to discretize the variational formulation, we modify the Stewart–Dellar formulation by introducing an explicit mapping between label spaces. Then all the integrations are over a single reference label space. Another addition to the variational formulation is that the multilayer shallow-water flow is dynamically coupled to the vessel motion. The theory will be developed in detail first for two-layers in Sect. 2 and then generalised to an arbitrary but finite number of layers in Sect. 4.

This system is a model for the Offshore Wave Energy Ltd (OWEL) ocean wave energy converter [4]. The OWEL wave energy converter (WEC) is essentially a rectangular box, open at one end to allow waves to enter and, once they have entered the device, the interior sloshing causes the wave to grow pushing air through a turbine and generating electricity. This interior system is a two-layer flow of air and water confined between upper and lower surfaces. This paper considers a simplified model of the OWEL configuration by including two-layers of differing density, but in a closed vessel. In Fig. 1, the vessel displacement *q*(*t*) could be prescribed, i.e. the vessel is subjected to a prescribed horizontal time-dependent force, or it could be determined as part of the solution. In this paper, we consider the vessel to be attached to a nonlinear spring, and hence, the vessel motion is governed by a combination of the restoring force of the spring and the hydrodynamic force of the fluid on the side walls of the vessel. The moving vessel walls in turn create a force on the fluid causing it to move, thus the system undergoes complex coupled motions.

The literature on two-layer flows in open systems, with and without a rigid lid is vast ([5, 6, 7, 8] to name a few), but in closed sloshing systems the literature is much more limited. The theoretical and experimental works of [9, 10] show excellent agreement for sloshing in a fixed rectangular tank with a rigid lid when a Lagrangian representation of the system is reduced to a system of ordinary differential equations with dissipative damping. Also, [11] examine two-layer sloshing in a forced vessel and derive a forced KdV equation when the layer thicknesses are comparable in size, an analysis of which shows that forcing induces chaos into the system through homoclinic tangles. The studies most relevant to the one in this paper examine the two-layer sloshing problem using a numerical scheme based upon a class of high resolution wave-propagating finite volume methods known as f-wave methods for both the forced [12] and the coupled problem [13]. This f-wave approach is very effective and can be readily be extended to multilayer systems [14] and systems with bottom topography [15], but [13] find the scheme is limited to layer density ratios of \(\rho _2/\rho _1\gtrsim 0.7\), where \(\rho _{1}\) and \(\rho _2\) are the fluid densities in the lower and upper layers, respectively, due to a linear growth in the system constraint error. Therefore this approach is not able to model the interior workings of the OWEL WEC, where the air/water density ratio is \(\rho _2/\rho _1\approx 10^{-3}\). The current paper formulates a simple numerical approach based upon a discretization of the LPP scheme, generalizing the numerics of [1] to two layers with nonlinear vessel motion. The LPP approach allows two-layer simulations with \(\rho _2/\rho _1=10^{-3}\) to be produced.

*i*satisfy

The paper is laid out as follows. In Sect. 2, the governing equations and variational principles for the two-layer rigid lid sloshing problem in the LPP description are presented. In Sect. 3 a variational discretization leading to a symplectic numerical integrator is introduced and simulations are presented. The results include validation of the scheme and extension of the numerical results into the non-Boussinesq regime. In Sect. 4, we demonstrate how the theory is extended to multilayer flows with a rigid upper lid and present simulations for the three-layer problem. Full details for the derivation of the governing three-layer equations is given in an appendix. Concluding remarks and discussion are presented in Sect. 5.

## 2 Coupled equations with a two-layer fluid

In this section, we develop the equations for two-layer sloshing in a vessel with rectangular cross-section with a rigid lid coupled to horizontal vessel motion. A schematic of the problem is shown in Fig. 1.

The special case of two-layer flow is of interest for two reasons: Firstly, to simplify the analysis and make the derivation of the governing equations and solution procedure tractable and readable, and secondly because the underlying motivation for this work comes from the two-layer air-water flow inside the OWEL WEC. In Sect. 4 we document how the method presented in this section can be extended to incorporate multilayer shallow-water flow, and present simulation results for the case of three layers.

*L*and height

*d*and we consider it filled with two immiscible, inviscid fluids of constant density \(\rho _1\) and \(\rho _2\) with \(\rho _1>\rho _2\). The problem is assumed to be two-dimensional with the effect of the front and back faces of the vessel neglected. In what follows, the subscripts 1 and 2 denote the lower- and upper-layer variables respectively. There are two frames of reference in this problem, the inertial frame with coordinates \(\mathbf{X}=(X,Y)\) and the body frame with coordinates \(\mathbf{x}_i=(x_i,y_i)\) in each layer \(i=1,2\). These coordinate systems are related via the time-dependent uniform translation

*q*(

*t*) in the \(x_1-\)direction, and in particular

*t*[13, 16]. The fluid in each layer must satisfy the no-penetration conditions on the vessel side walls, and hence, the boundary conditions are

*a priori*and is determined by a combination of a restoring force, such as a spring or a pendulum [17] and a hydrodynamic force exerted on the vessel side walls by the sloshing fluid. We assume that the vessel is connected to the spatial origin by a nonlinear spring, and hence, the vessel motion is governed by

*Eulerian variational principle*by considering variations to the Lagrangian functional

*t*, but the integral, as written above, is over \(x_1\), moreover, as discussed earlier, the Eulerian constraint \(h_1(x_1,t)+h_2(x_2,t)=d\) has to hold for \(x_1=x_2\). Both of these issues are overcome in Sect. 2.1 by introducing the constraint that \(x_1=x_2\) into (2.7) and formulating the problem in terms of the lower layer coordinate only.

The shallow-water equations (2.2)–(2.5) could be solved numerically via some implicit shallow-water numerical scheme, with the vessel equation (2.6) integrated via standard fourth-order Runge–Kutta integration. However, this scheme would not necessarily have good energy conservation properties. Hence, in order to model the long-time oscillatory behaviour of the system, we construct a Hamiltonian formulation of the system in order to utilise geometric integration schemes. We do this by transforming the above Eulerian variational formulation to an LPP Lagrangian variational formulation.

### 2.1 LPP description

#### 2.1.1 Lagrangian variational formulation

To transform the Eulerian shallow-water equations into a LPP formulation, we need to consider mappings from the Lagrangian particle labels and Lagrangian time \((a_i,\tau )\) in each layer to the corresponding Cartesian coordinates and Eulerian time \((x_i,t)\). This again demonstrates another peculiar feature of the problem, because there is no guarantee that for all \(\tau \), \(x_1(a_1,\tau )=x_2(a_2,\tau )\) which we require so as to satisfy the Eulerian constraint (2.1). The approach to overcome this problem is to link the two LPP labels in each layer via \(a_2=\phi (a_1,\tau )\) where \(\phi (a_1,\tau )\) is an unknown map to be determined. In the subsequent analysis, we shall drop the subscript 1 from the Lagrangian label \(a_1\) with the understanding that this is the label in the lower layer, and our primary reference label.

*a*and \(\tau \) denote partial derivatives. Because we have assumed the constraint (2.9) the derivatives in (2.4) and (2.5) map in the same way as in (2.10), but we can show this formally. From the form of \(x_2\) in (2.9), the derivatives in the LPP setting map on to

*q*, which takes the form

*q*(e.g. writing \(q=q+\delta q\) with \(\delta q(\tau _1)=\delta q(\tau _2)=0\)) leads to (2.17) and (2.18) respectively.

Note that in the case \(\rho _2=0\) (with \(\nu _2=0\)), (2.19) reduces to the one-layer coupled Lagrangian given in [1], i.e. in this case the fluid does not feel the effect of the rigid lid.

#### 2.1.2 Hamiltonian formulation

The form of (2.23) is equivalent to that in (2.17), which was derived directly from the Eulerian form of the equations. This equivalence is shown in Appendix 1.

### 2.2 Linear solutions to LPP problem

The linear solution of the two-layer shallow-water sloshing problem with a rigid lid in the Eulerian framework has been discussed in detail in [13]. However, the form of this linear solution in the LPP framework would be desirable in order to use it as an initial condition when numerically integrating Hamilton’s equations so to validate the scheme. Hence, we briefly outline the linear solution procedure here.

*s*is found from (2.33) then \(\omega \) is given by

## 3 Variational discretization and computation

### 3.1 Numerical algorithm

*N*parcels by setting

*x*here) and \(w_i(t):=w(a_i,\tau )\). The derivatives are discretized using forward differences, except when \(i=N+1\) where backward differences are used, and the integrals are approximated using the trapezoidal rule.

Equations (2.24)–(2.26) can be discretized in a straightforward manner, as the variables for which variations are taken, do not appear differentiated with respect to *a* on the RHS of the equations. However, in (2.23), derivatives of \(x_{1}\) with respect to *a* do appear in the RHS, and thus, it is not clear how to discretize these equations. To overcome this, we use a semi-discretization of the Hamiltonian, where the Hamiltonian is discretized and then variations with respect to \(x_i\) are taken.

*w*or 1, and therefore it can be shown that

*N*equations for the \(2N+4\) unknowns. The remaining 4 equations come from the boundary conditions

*implicit-midpoint rule*approach. In this case, the system becomes the set of nonlinear algebraic equations

*n*denotes the time-step, such that \(\mathbf{p}^n=\mathbf{p}(n\Delta \tau )\). This system of implicit equations are solved at each new time step via Newton iterations. In order to increase the speed of the iteration scheme, the method of [18] is employed to iteratively calculate the inverse Jacobian matrix after the first iteration of the first time step.

### 3.2 Numerical results

Parameter (units) | |||||
---|---|---|---|---|---|

| 1 | 1 | 1 | 1 | 1 |

| 0.12 | 0.12 | 0.08 | 0.12 | 0.12 |

\(h_1^0\) (m) | 0.06 | 0.06 | 0.04 | 0.08 | 0.04 |

\(h_2^0\) (m) | 0.06 | 0.06 | 0.04 | 0.04 | 0.04 |

\(h_3^0\) (m) | – | – | – | – | 0.04 |

\(\rho _1\) (\(\mathrm{kg\,m}^{-3}\)) | 1000 | 1000 | 1000 | 1000 | 1000 |

\(\rho _2\) (\(\mathrm{kg\,m}^{-3}\)) | 900 | 1 | 700 | 1 | 500 |

\(\rho _3\) (\(\mathrm{kg\,m}^{-3}\)) | – | – | – | – | 1 |

\(m_f^{(1)}\) (kg) | 60 | 60 | 40 | 80 | 40 |

\(m_f^{(2)}\) (kg) | 54 | 0.06 | 28 | 0.04 | 20 |

\(m_f^{(3)}\) (kg) | – | – | – | – | 0.04 |

\(m_v\) (kg) | 10 | 10 | 3.4 | 10 | 10 |

\(\widehat{q}\) (m) | \(1\times 10^{-4}\) | \(1\times 10^{-4}\) | 0.01 | 0.07 | 0.07 |

\(\widehat{q}_1\) (m) | \(1\times 10^{-4}\) | \(1\times 10^{-4}\) | 0 | 0 | – |

\(\widehat{q}_2\) (m) | \(1\times 10^{-4}\) | \(1\times 10^{-4}\) | 0.1 | 0 | – |

\(\nu _1\) (\(\mathrm{kg\,s}^{-2}\)) | 100 | 100 | 189.40 | 100 | 100 |

\(\nu _2\) (\(\mathrm{kg\,m}^{-2}\mathrm{s}^{-2}\)) | 0 | 0 | –189.40 | 800 | 800 |

\(\omega \) (\(\mathrm{s^{-1}}\)) | 0.8995 | 1.0980 | – | – | – |

*q*(

*t*) and the surface interface evolution \(h_1(x_1,t)\) along with time evolutions of the total vessel energy \(E_v(t)\) and the total fluid energy \(E_f(t)\) defined by

The density ratio \(\rho _2/\rho _1=0.7\) in Figs. 6 and 7 is on the borderline between the Boussinesq and non-Boussinesq regimes. The f-wave numerical scheme developed by [13] works most effectively in the Boussinesq regime, especially for weakly nonlinear simulations. The scheme encounters problems satisfying the system constraints for density ratios \(\rho _1/\rho _1\lesssim 0.7\). The Hamiltonian scheme developed here has the rigid lid and mass-flux conditions (2.1) and (2.16) directly built in to the scheme and so is able to resolve simulations for for these density ratios. Figs. 8 and 9 show results for \(\rho _2/\rho _1=10^{-3}\), which is the density ratio of air to water for an initial condition akin to those found in an experimental setup, \(\widehat{q}_1=\widehat{q}_2=0\). As the initial interface is flat, the initial condition consists of an infinite sum of all the sloshing modes in (2.33) at different amplitudes, and thus, the result is the lowest frequency mode superposed with higher frequency modes, giving the small oscillations on the results. The energy error \(\widehat{\mathscr {H}}_N(t)\) in Fig. 8b, although larger than the result in Fig. 6b, is still relatively small \(O(10^{-5})\), and bounded for the time-scale of the simulations. The results in Fig. 9 depict the interface gently sloshing back and forth in the vessel, and as it does so it becomes increasingly more fine scaled. This is a well known characteristic when symplectic schemes are applied to sloshing problems [19] and is due to the energy of the system cascading down to the high frequency modes, in what is essentially a spectral scheme. However, as the numerical time integrator is symplectic, it conserves this energy and so this energy remains in the high frequency modes as these high frequency oscillations. These could be removed using an artificial viscosity term or the filtering scheme used by [20], but the numerical scheme will then no longer be energy conserving.

The two-layer results presented here show the capabilities of the Hamiltonian approach for these multilayer sloshing problems. Note, however, despite the introduction of the mapping \(\phi (a_1,\tau )\) to ensure \(x_1(a_1,\tau )=x_2(\phi (a_1,\tau ),\tau )\), this mapping was never discussed or plotted. This is because the two-layer problem is in fact a special case of the multilayer sloshing problem, because equations (2.1) and (2.16) mean that the upper-layer variables can be eliminated and the problem can be formulated solely in terms of lower-layer variables. In the next section, we formulate the general *M*-layer shallow-water problem, and present results for three-layer sloshing, where the mappings \(\phi _i\) do need to be calculated.

## 4 Extension to multilayer shallow-water flows

*M*-layer shallow-water problem is straightforward, with the biggest difference being the necessity to calculate the mapping functions \(\phi _i(a_1,\tau )\). The derivation and analysis can get a bit lengthy so detail is recorded in Appendix 3 for the three-layer case. A schematic of the general

*M*-layer problem is shown in Fig. 10. Here we will impose the constraint \(x_1=x_2=...=x_M\) from the outset in order to simplify the analysis.

*M*-layer shallow-water equations with a rigid lid are

*M*. Now introducing the LPP mapping (2.8) into the layer 1 mass conservation equation leads again to (2.11) and hence (2.15). Thus, \(\widehat{h}_1\) is now written in terms of \(x_1\) only, with \(u_1=x_{1\tau }\) its associated momenta. However, unlike the two-layer case, we still have layer variables \((h_2,u_2),\ldots ,(h_{M-1},u_{M-1})\) to eliminate from the Lagrangian and replace by some position variable and its associated momenta.

### 4.1 Numerical implementation for three layers

*M*-layer problem, we present a result for coupled three-layer sloshing in Figs. 11 and 12. Details of the derivation of the three-layer Hamiltonian and symplectic integration scheme, as well as validation of the scheme, are given in Appendix 3. The initial conditions for these simulations are

The results in Figs. 11 and 12 can be directly compared with those in Figs. 8 and 9, as they are essentially the same parameter values except for the inclusion of a third, less-dense, middle layer. These results show a vessel motion whose amplitude is strongly modulated by the inclusion of the third layer. This modulated vessel displacement is due to the hydrodynamic force on the vessel walls slowly becoming out of phase with the restoring force of the spring, before slowly moving back in phase. This more complex behaviour is not a surprise as the characteristic equation for this system (8.40) has more solutions compared to the two-layer equation (2.33) due to the inclusion of the additional interface. The interface profiles again show fine scale structure at later times, but at \(t=29\) there exists fairly large oscillations at the lower interface. Also, the energy error \(\widehat{\mathscr {H}}_N(t)\) in Fig. 11b, while still small, \(O(10^{-5})\), grows moderately over the time frame of the simulation. The reason for these two observations, we believe, is due to the Kelvin–Helmholtz instability on the interface [21], and we use a smaller time-step to stop the error growing more rapidly. This is more evident in the validation simulation in Appendix 3. Hence, one has to check the energy error \(\widehat{\mathscr {H}}_N(t)\) for a calculation to determine whether it is still within tolerable bounds. Again the introduction of artificial viscosity or filtering would help limit this instability by removing the fine-scale high-frequency modes from the system, which grow fastest in an inviscid system [22].

## 5 Conclusions and discussion

This paper documents the Lagrangian variational formulation of the LPP representation of multilayer shallow-water sloshing, coupled to horizontal vessel motion governed by a nonlinear spring. The Lagrangian variational formulation was transformed to a Hamiltonian formulation which has nice properties for numerical simulation. A symplectic numerical integration scheme was applied to the resulting set of Hamiltonian partial differential equations for the two-layer problem, and results of the simulations were found to be in excellent agreement with the linear solution and the nonlinear results of the f-wave scheme of [13]. Using this Hamiltonian formulation the results of [13] were extended into the non-Boussinesq regime, with a result presented for a density ratio \(\rho _2/\rho _1=10^{-3}\), akin to that of air over water.

The Hamiltonian formulation was presented in detail for the two-fluid system, but the solution procedure was generalised in Sect. 4 to a system of *M*-fluid layers coupled to horizontal vessel motion where the vessel is attached to a nonlinear spring. Results were presented for a three-layer system, with the full derivation confined to Appendix 3. Results for the three-layer system showed a system energy error which grew slowly in time, due to the Kelvin-Helmholtz instability on the fluid interfaces. For the results presented in this paper, this error growth was small and thus tolerable for the time frame of the simulations. However, this error would need to be examined in fully nonlinear simulations or long-time simulations to make sure it was small compared to the fluid velocities and vessel displacement. Also, in thin layers, where fluid velocities tend to be larger to conserve the mass flux (4.6), this instability could be an issue. Surface tension or a filter could be added to mollify the instability.

As this work was motivated by studying the WEC of Offshore Wave Energy Ltd (OWEL), a direction of great interest is to extend the vessel motion to incorporate rotation (pitch) along with the translations considered here, and to incorporate influx-efflux boundary conditions at the side walls, which model the waves entering the device and leaving through the extraction route. In the OWEL WEC, the wetting and drying of the upper rigid lid is very important for the modelling of the power-take-off mechanism. The current two-layer approach considered in this paper cannot incorporate this phenomena. The reason for this comes from the mass-flux equation \(h_1u_1+h_2u_2=0\) which holds throughout the fluid. We find that as \(h_2\rightarrow 0\) in this expression, despite the momentum \(h_2u_2\) reducing in size, the value of \(u_2\) becomes large which causes numerical difficulties in the current scheme. Thus, another area of great interest is to incorporate this feature into the model.

## Notes

### Acknowledgements

This work is supported by the EPSRC under Grant number EP/K008188/1. Due to confidentiality agreements with research collaborators, supporting data can only be made available to bona fide researchers subject to a non-disclosure agreement. Details of the data and how to request access are available from the University of Surrey publications repository: researchdata@surrey.ac.uk

## References

- 1.Alemi Ardakani H, Bridges TJ (2010) Dynamic coupling between shallow-water sloshing and horizontal vehicle motion. Eur J Appl Math 21:479–517MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Alemi Ardakani H (2016) A symplectic integrator for dynamic coupling between nonlinear vessel motion with variable cross-section and bottom topography and interior shallow-water sloshing. J Fluids Struct 65:30–43CrossRefGoogle Scholar
- 3.Stewart AL, Dellar PJ (2010) Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J Fluid Mech 651:387–413ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 4.Leybourne M, Batten W, Bahaj AS, O’Nians J, Minns N (2010) Experimental and computational modelling of the OWEL wave energy converter. In: 3rd international conference on ocean energy, Bilbao, SpainGoogle Scholar
- 5.Ovsyannikov LV (1979) Two-layer shallow water model. J Appl Mech Tech Phys 20:127–135ADSCrossRefGoogle Scholar
- 6.Lawrence GA (1990) On the hydraulics of Boussinesq and non-Boussinesq two-layer flows. J Fluid Mech 215:457–480ADSCrossRefzbMATHGoogle Scholar
- 7.Bridges TJ, Donaldson NM (2007) Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves. Phys Fluids 19:072111ADSCrossRefzbMATHGoogle Scholar
- 8.Kim J, LeVeque RJ (2008) Two-layer shallow water system and its applications. In: Proceedings of the twelfth international conference on hyperbolic problems, College ParkGoogle Scholar
- 9.La Rocca M, Sciortino G, Boniforti MA (2002) Interfacial gravity waves in a two-fluid system. Fluid Dyn Res 30:31–66ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 10.La Rocca M, Sciortino G, Adduce C, Boniforti MA (2005) Experimental and theoretical investigation on the sloshing of a two-liquid system with free surface. Phys Fluids 17(6):062101ADSCrossRefzbMATHGoogle Scholar
- 11.Mackey D, Cox EA (2003) Dynamics of a two-layer fluid sloshing problem. IMA J Appl Math 68(6):665–686MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Alemi Ardakani H, Bridges TJ, Turner MR (2016) Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid. J Comput Appl Math 296:462–479MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Alemi Ardakani H, Bridges TJ, Turner MR (2015) Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing. J Fluid Struct 59:432–460CrossRefGoogle Scholar
- 14.Mandli KT (2011) Finite volume methods for the multilayer shallow water equations with applications to storm surges. PhD thesis, University of WashingtonGoogle Scholar
- 15.George DL (2006) Finite volume methods and adaptive refinement for tsunami propagation and inundation. PhD thesis, University of WashingtonGoogle Scholar
- 16.Baines PG (1998) Topographic effects in stratified flows. Cambridge University Press, CambridgezbMATHGoogle Scholar
- 17.Alemi Ardakani H, Bridges TJ, Turner MR (2012) Resonance in a model for Cooker’s sloshing experiment. Eur J Mech B 36:25–38MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Broyden CG (1965) A class of methods for solving nonlinear simultaneous equations. Math Comput 19(92):577–593MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Alemi Ardakani H, Bridges TJ (2012) Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in two dimensions. Eur J Mech B 31:30–43MathSciNetCrossRefzbMATHGoogle Scholar
- 20.Alemi Ardakani H, Turner MR (2016) Numerical simulations of dynamic coupling between shallow-water sloshing and horizontal vessel motion with baffles. Fluid Dyn Res 48(3):035504ADSMathSciNetCrossRefGoogle Scholar
- 21.Drazin PG (2002) Introduction to hydrodynamic stability. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- 22.Drazin PG, Reid WH (2004) Hydrodynamic stability. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- 23.Johnson RS (1997) A modern introduction to the mathematical theory of water waves. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar

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