Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions

Transcription

1 Under consideration for publication in J. Fluid Mec. 1 Sallow-water slosing in vessels undergoing prescribed rigid-body motion in tree dimensions (draft version wit colour figures) H A M I D A L E M I A R D A K A N I A N D T H O M A S J. B R I D G E S Department of Matematics, University of Surrey, Guildford, Surrey GU2 7XH England (Received?? and in revised form??) New sallow-water equations, for slosing in tree dimensions (two orizontal and one vertical) in a vessel wic is undergoing rigid-body motion in 3 space, are derived. Te rigid-body motion of te vessel (roll-pitc-yaw and/or surge-sway-eave) is modelled exactly and te only approximations are in te fluid motion. Te flow is assumed to be inviscid but vortical, wit approximations on te vertical velocity and acceleration at te surface. Tese equations improve previous sallow water models. Te model also extends te essence of te Penney-Price-Taylor teory for te igest standing wave. Te surface sallow water equations are simulated using an split-step implicit alternating direction finitedifference sceme. Numerical experiments are reported, including comparisons wit existing results in te literature, and simulations wit vessels undergoing full tree-dimensional rotations. 1. Introduction Sallow-water equations (SWEs) for a tree-dimensional (3D) inviscid but vortical fluid in a vessel undergoing an arbitrary prescribed rigid-body motion in 3D are derived. Te rigid body motion is represented exactly and only two assumptions are imposed on te velocity and acceleration at te free surface to close te SWEs. Wile tere as been a vast amount of researc into two-dimensional slosing, te researc into 3D slosing is still very muc in development. A review of muc of te researc to date is presented in Ibraim (2005). Te predominant teoretical approaces for studying 3D slosing are (a) asymptotics and weakly nonlinear teories, (b) multi-modal expansions wic reduce te governing equations to a set of ordinary differential equations; (c) reduction to model partial differential equations suc as te sallow water equations; and (d) direct numerical simulation of te full 3D problem. If 3D numerical simulations were faster, te latter approac would be very appealing. Tere as been muc progress in te numerical simulation of 3D slosing using Navier-Stokes based metods (MAC, SURF, VOF, RANSE), boundary-element metods and finite-element metods for 3D potential flow (some examples are Lee et al. 2007; Kim 2001; Liu & Lin 2008; Bucner 2002; Kleefsman et al. 2005; Gerrits 2001; aus der Wiesce 2003; Cen et al. 2009; Cariou & Casella 1999; Cen et al. 2000; Wu et al. 1998). Wile te results of tese simulations are impressive, te difficulty is tat CPU times are measured in ours rater tan seconds or minutes. An example is te VOF simulations of Liu & Lin (2008), were 3D slosing in a vessel wit rectangular base is forced armonically. To simulate 50 seconds of real time (about 20 periods of armonic forcing) took 265 ours of CPU time. It is very difficult to do parametric studies or long time simulations wit tis amount of CPU time. Hence any reduction in dimension is appealing. On te oter and one can make some progress in te understanding of slosing in 3D using analytical metods, asymptotics (perturbation teory, multi-scale expansions) and modal expansions. In 2D sallow water slosing te predominant types of solution are te standing wave and travelling ydraulic jump. But in 3D sallow water slosing te range of basic solutions is muc larger. One still as te 2D solutions, but tere can be mixed modes, swirling modes, multi-mode cnoidal standing waves and multi-dimensional ydraulic jumps and analytical metods are very effective for identifying parameter regimes for tese basic solutions (e.g. Faltinsen et al. 2003, 2006a; Faltinsen & Timoka

2 2 H. Alemi Ardakani and T. J. Bridges 2003; Bridges 1987). And, tere is a vast literature on asymptotic metods for te special case of parametrically-forced slosing in rectangular containers (Faraday experiment) (e.g. Miles & Henderson 1990, and its citation trail). Multi-modal expansions take analytic metods to iger order. Wen te fluid domain is finite in extent tere is a countable basis of eigenfunctions for te basin sape, and one approac tat as been extensively used is to expand te nonlinear equations in terms of tese (or oter) basis functions wit time-dependent coefficients, leading to a large system of ordinary differential equations. Examples of tis approac are Faltinsen & Timoka (2003); Faltinsen et al. (2003, 2006a); La Rocca et al. (1997, 2000) and (Part II of Ibraim 2005). Te advantage of modal approximation is tat te problem is reduced to a system of ODEs wic is muc quicker to simulate numerically. In between direct 3D simulations and analytical metods is a tird approac: to derive reduced PDE models. Tis approac is particularly useful for te study of sallow water slosing. Te study of 3D sallow-water slosing is motivated by a number of applications suc as: slosing on te deck of fising vessels (Caglayan & Storc 1982; Adee & Caglayan 1982) and offsore suppy vessels (Falzarano et al. 2002), slosing in wing fuel tanks of aircraft (Disimile et al. 2009), green-water effects (Dillingam & Falzarano 1986; Zou et al. 1999; Bucner 2002; Kleefsman et al. 2005), slosing in a swimming pool on deck (Ruponen et al. 2009), slosing in fis tanks onboard fising vessels (Lee et al. 2005), and slosing in automobile fuel tanks (aus der Wiesce 2003). About te same time, Dillingam & Falzarano (1986) and Pantazopoulos (1987) (see also Pantazopoulos & Adee (1987) and Pantazopoulos (1988)) derived sallow water equations for 3D slosing wit te vessel motion prescribed. It is tis approac tat is te starting point for te current paper. Tese SWEs, ereafter called te DFP SWEs, will be recorded and analyzed in 9. In followup work Huang (1995) (see also Huang & Hsiung (1996) and Huang & Hsiung (1997)) gave an alternative derivation of rotating SWEs for slosing resulting in sligtly different equations from te DFP SWEs. Te latter system will be called te HH SWEs. A discussion of te HH SWEs is also given in 9. Te DFP SWEs use a very simple form for te vessel motion, and ave unecessarily restrictive assumptions in te derivation. Te derivation of te HH SWEs is more precise but still as some restrictive assumptions. Tese assumptions are outlined in 9 and in more detail in te tecnical reports of Alemi Ardakani & Bridges (2009d,e). In tis paper a new derivation of SWEs in 3D is given wic leads to exact SWEs for te orizontal velocity field at te free surface of te form ( U t + UU x + V U y + a 11 + Dw ) x + a 12 y = b 1 + σ x div(κ) ( V t + UV x + V V y + a 22 + Dw ) y + a 21 x = b 2 + σ y div(κ). Tese equations are relative to a frame of reference moving wit te vessel wit coordinates (x, y, z) and z vertical. Te terms a 11, a 12, a 21, a 22, b 1, b 2 encode te moving frame, κ is a curvature term associated wit surface tension, and Dw is te Lagrangian vertical acceleration at te free surface. Te fluid occupies a rectangular region wit a single-valued free surface, Te free surface orizontal velocity field is 0 z (x, y, t), 0 x L 1, 0 y L 2. U(x, y, t) = u(x, y, z, t) := u(x, y, (x, y, t), t) and V (x, y, t) = v(x, y, z, t). Couple te equations (1.1) wit te exact mass conservation equation (1.1) t + (U) x + (V ) x = W + U x + V y, (1.2) wic is derived from te kinematic free surface boundary condition (see 2). W = w is te vertical velocity at te free surface. By assuming tat Dw 0 and W + U x + V y 0 te equations (1.1)-(1.2) are a closed set of SWEs wic retain te vessel motion exactly. It is tis closed set of SWEs tat is te starting point for te analysis and numerics in tis paper.

3 Sallow water slosing in 3D 3 One of te advantages of te SWEs is tat vorticity is retained. Tis is in contrast to almost all analytical researc into 3D slosing wic uses te assumption of irrotationality. Vorticity can be input troug te initial conditions, but a new mecanism comes into play in sallow water slosing: te creation of vorticity toug discontinuities in ydraulic jumps (e.g. Peregrine 1998, 1999). In coastal ydraulics Peregrine (1998) as sown ow steady ydraulic jumps generate vorticity. In te case of sallow water slosing tere is a dynamic mecanism due to te time-dependent nature of te ydraulic jumps, and tis penomena is witnessed in some of te numerical simulations reported erein. In special cases te surface SWEs (1.1)-(1.2) ave a form of potential vorticity (PV) conservation. For example wen te vessel is undergoing pure yaw motion, ten a form of PV is conserved. Tis PV is different from tat appearing in geopysical fluid dynamics. In geopysical fluid dynamics te rotation (rotation of te eart) is treated as a constant, wereas ere te rotation is time dependent. On te oter and, Barnes et al. (1983) ave sown tat te eart does indeed wobble and so te idea of time-dependent yaw motion as some relation to geopysical fluid dynamics. Te vessel is modelled as a rigid body, and te position of a rigid body in 3 space is completely determined by specifying (q(t), Q(t)) were q(t) is a vector in R 3 giving te orizontal and vertical translation of te body relative to some fixed reference frame, and Q is a proper 3 3 rotation matrix (Q is ortogonal and det(q) = 1). Specifying translations is straigtfoward, but specifying rotations requires a little more care. Surprisingly most previous work on forced slosing uses pure translation, or te rotations are simplified using a small angle approximation or restricted to planar rotations. Te small angle approximation is to take te angular velocity of te form Ω = ( φ, θ, ψ) were φ, θ and ψ are roll, pitc and yaw angles respectively (precise definition given in 8.2). Examples of forcing used in te literature are Cen et al. (2009) (armonic surge and sway motion, and armonic roll motion); Wu et al. (1998) (armonic surge, sway and eave forcing); Cen et al. (2000) and Faltinsen et al. (2006a) (armonic surge forcing); aus der Wiesce (2003) (impulse excitation representative of automobile accelerations); Faltinsen et al. (2006b) (roll-pitc forcing wit a small angle approximation); Liu & Lin (2008) and Wu & Cen (2009) include forcing in all 6 degrees of freedom, but te rotations use te small angle approximation. In tis paper exact representations of te rotations are used. Bot Euler angle representations and numerical construction of te rotation matrix are used. Te coice (body or space) representation of te angular velocity is important and its implications are discussed. Also teir are subtleties in te construction of te angular velocity (Leubner 1981) and tese are also discussed erein and in te report of Alemi Ardakani & Bridges (2009f ). Tere as been very little experimental work wit vessels undergoing full 3D rotations. Most experiments are wit pure translations and/or planar rotations. A facility for 3D rotations of vessels wit fluid would be tecnically demanding. However, te paper of Disimile et al. (2009) mentions an experimental facility capable of exciting a tank containing fluid in all 6 degrees of freedom. However, to date tey ave only reported on results of forced roll motion. Our principal tool for analyzing te SWEs is numerics. Te numerical sceme is based on te Abbott-Ionescu sceme wic is widely used in computational ydraulics. It is a finite-difference sceme, fully implicit, and te two-dimensionality is treated using an alternating direction implicit sceme. A one-dimensional version of tis sceme was used in Alemi Ardakani & Bridges (2009g). An overview of te paper is as follows. In 3-4 te SWEs (1.1) for slosing in a vessel undergoing motion in 3-space are derived starting from te full 3D Euler equations relative to a moving frame. Before assuming tat te Lagrangian vertical accelerations are small we analyze te exact equations in 5 and sow tat tey give an explanation for and a generalization to 3D of te Penney-Price-Taylor teory for te igest standing wave. Te assumptions necessary for te reduction to a closed set of SWEs wit te body motion exact are discussed in 7, leading to a closed set of SWEs. Details of te specification of te vessel motion are given in 8, including bot an Euler angle representation and direction calculation of te rotation matrix numerically. Furter detail on te special case of yaw-pitc-roll Euler angles is given in te report of Alemi Ardakani & Bridges (2009c).

4 4 H. Alemi Ardakani and T. J. Bridges z x d Z y X Y Figure 1. Scematic sowing a configuration of te fixed coordinate system OXY Z relative to te moving coordinate system, attaced to te tank, Oxyz. Te origin of te OXYZ coordinate system can ave an additional displacement q(t). In tis figure q(t) = 0. Te DFP SWEs and HH SWEs are reviewed in te reports of Alemi Ardakani & Bridges (2009d) and Alemi Ardakani & Bridges (2009e). A summary of te main assumptions in teir derivation and a comparison wit te new surface SWEs is given in 9. Te numerical algoritm is introduced in 10. Furter detail on te numerical sceme is contained in te tecnical report of Alemi Ardakani & Bridges (2009f ). Numerical results are reported in Sections 11 to 14. Tese simulations include full 3D rotations: roll-pitc in 11 and yaw-pitc-roll in 14. Results of pure yaw forcing are report in 12. Tese simulations sow te appearance of vorticity, agree wit previous simulations of Huang & Hsiung (1996) and conserve a time-dependent form of potential vorticity. Motivated by simulations of Wu & Cen (2009), diagonal surge-sway forcing is considered in 13, illustrating planarization of waves and te appearance of square-like waves. 2. Governing equations Te configuration of te fluid in a rotating-translating vessel is sown scematically in Figure 1. Te vessel is a rigid body wic is free to rotate and or translate in R 3, and tis motion will be specified. Te spatial frame, wic is fixed in space, as coordinates denoted by X = (X, Y, Z), and te body frame a moving frame is attaced to te vessel and as coordinates denoted by x = (x, y, z). Te wole system is translating in space wit translation vector q(t). Te position of a particle in te body frame is terefore related to a point in te spatial frame by X = Q(x + d) + q, were Q is a proper rotation in R 3 (Q T = Q 1 and det(q) = 1). Te axis of rotation can be displaced a distance d from te origin of te body frame and d = (d 1, d 2, d 3 ) R 3 is constant. Te displacement Qd could be incorporated into q(t) but in cases were te origin of te spatial frame is fixed, it will be useful to maintain te distinction. Tis formulation is consistent wit te teory of rigid body motion, were an arbitrary motion

5 Sallow water slosing in 3D 5 can be described by te pair (Q(t), q(t)) wit Q(t) a proper rotation matrix and q(t) a vector in R 3 (O Reilly 2008; Murray et al. 1994). Te body angular velocity is a time-dependent vector Ω(t) = (Ω 1 (t), Ω 2 (t), Ω 3 (t)), wit entries determined from Q by Q T Q = 0 Ω 3 Ω 2 Ω 3 0 Ω 1 := Ω. (2.1) Ω 2 Ω 1 0 Te convention for te entries of te skew-symmetric matrix Ω is suc tat Ωr = Ω r, for any r R 3, Ω := (Ω 1, Ω 2, Ω 3 ). Te body angular velocity is to be contrasted wit te spatial angular velocity te angular velocity viewed from te spatial frame wic is Ω spatial := QQ T. As vectors te spatial and body angular velocities are related by Ω spatial = QΩ. Eiter representation for te angular velocity can be used. For example Pantazopoulos (1988); Dillingam & Falzarano (1986); Falzarano et al. (2002); Pantazopoulos & Adee (1987) all use te spatial representation, wereas Huang & Hsiung (1996, 1997) use te body representation. We will sow tat te body representation is te sensible coice leading to great simplification of te equations. Hencefort, te angular velocity witout a superannotation will represent te body angular velocity. Te velocity and acceleration in te spatial frame are and Ẋ = Q(ẋ + Ω (x + d)) + q. (2.2) Ẍ = Q[ẍ + 2Ω ẋ + Ω (x + d) + Ω Ω (x + d) + Q T q]. (2.3) Newton s law is expressed relative to te spatial frame, but substitution of (2.2)-(2.3) into Newton s law and multiplying by Q T gives te governing equations relative to te body frame, Du + 1 ρ p + 2Ω u + Ω (x + d) + Ω (Ω (x + d)) + Q T g + Q T q = 0, (2.4) were u = (u, v, w) is te velocity field, D := t + u x + v y + w z and 0 g := g 0, 1 wit g > 0 te gravitational constant. A detailed derivation is given in Appendix A of Alemi Ardakani & Bridges (2009g). Te term Q T g rotates te usual gravity vector so tat its direction is properly viewed in te body frame. Similarly for te translational acceleration q. Furter comments on te viewpoint of te vessel velocity and acceleration are in 8. In te derivation of te SWEs te components of (2.4) will be needed. Write te momentum equation as Du + 1 p = F, (2.5) ρ ten F = 2Ω u Ω (x + d) Ω (Ω (x + d)) Q T g Q T q,

6 6 H. Alemi Ardakani and T. J. Bridges and te components of F are Te fluid occupies te region F 1 = 2(Ω 2 w Ω 3 v) Ω 2 (z + d 3 ) + Ω 3 (y + d 2 ) Ω 1 Ω (x + d) + (x + d 1 ) Ω 2 q Qe 1 ge 3 Qe 1 F 2 = +2(Ω 1 w Ω 3 u) + Ω 1 (z + d 3 ) Ω 3 (x + d 1 ) Ω 2 Ω (x + d) + (y + d 2 ) Ω 2 q Qe 2 ge 3 Qe 2 F 3 = 2(Ω 1 v Ω 2 u) Ω 1 (y + d 2 ) + Ω 2 (x + d 1 ) Ω 3 Ω (x + d) + (z + d 3 ) Ω 2 q Qe 3 ge 3 Qe 3. 0 x L 1, 0 y L 2, 0 z (x, y, t), were te lengts L 1 and L 2 are given positive constants, and z = (x, y, t) is te position of te free surface. Conservation of mass relative to te body frame takes te usual form Te boundary conditions are u x + v y + w z = 0. (2.6) u = 0 at x = 0 and x = L 1 v = 0 at y = 0 and y = L 2 w = 0 at z = 0, and at te free surface te boundary conditions are te kinematic condition and te dynamic condition were σ > 0 is te coefficient of surface tension, (2.7) t + u x + v y = w, at y = (x, t), (2.8) p = ρσ div(κ) at y = (x, t), (2.9) and κ 1 = x x + 2 y div(κ) = κ 1 x + κ 2 y, and κ 1 = y x + 2 y. Te vorticity vector is defined by Differentiating tis equation gives 2.1. Vorticity V := u, DV = V u + Taking te curl of te momentum equations (2.4) gives ( ) Du = 2Ω u 2 Ω. Combining wit (2.10) gives te vorticity equation DV = (2Ω + V) u 2 Ω. ( ) Du. (2.10)

7 Sallow water slosing in 3D 7 Two components of te vorticity equation will be important in te derivation of te surface SWEs, ( ) Dw = ( ) Dv u + 2Ω 1 y z x + 2Ω u 2 y + 2Ω u 3 z 2 Ω 1 ( ) Dw = ( ) (2.11) Dv v 2Ω 1 x z x 2Ω v 2 y 2Ω v 3 z + 2 Ω Reduction of te pressure gradient Te derivation of te SWEs in 3D (1.1) follows te same strategy as te 2D case in Alemi Ardakani & Bridges (2009g). Te key is te precise treatment of te pressure field. Let β(x, y, t) = Ω 1 (y + d 2 ) + Ω 2 (x + d 1 ) Ω 3 Ω 1 (x + d 1 ) Ω 3 Ω 2 (y + d 2 ) Qe 3 q gqe 3 e 3. (3.1) Ten te vertical momentum equation can be expressed in te form Integrate from z to, Dw ds + 1 ρ p z Dw + 1 p ρ z = 2(Ω 1v Ω 2 u) + (Ω Ω 2 2)(z + d 3 ) + β(x, y, t). z = 2Ω 1 v ds + 2Ω 2 u ds z z +(Ω Ω 2 2)( z2 + d 3 d 3 z) + β(x, y, t)( z). Applying te surface boundary condition on te pressure ten gives te pressure at any point z, 1 p(x, y, z, t) = ρ z Dw ds + 2Ω 1 v ds 2Ω 2 u ds β(x, y, t)( z) z z (Ω Ω 2 2)( z2 + d 3 d 3 z) σdiv(κ). Tis equation for p(x, y, z, t) is exact. Te strategy is to take derivatives wit respect to x and y and ten substitute into te orizontal momentum equations. Te details are lengty and are given in Appendix A. Te expressions are and 1 p ρ x = Du 1 p ρ y z + Dw x + 2Ω 2 W 2Ω 3 V + 2 Ω 2 ( z) 2Ω 2 w + 2Ω 3 v +2Ω 1 V x 2Ω 2 U x (Ω Ω 2 2)( + d 3 ) x β x ( z) β x σ x div(κ). = Dv z + Dw y 2Ω 1 W + 2Ω 3 U 2 Ω 1 ( z) + 2Ω 1 w 2Ω 3 u +2Ω 1 V y 2Ω 2 U y (Ω Ω 2 2)( + d 3 ) y β y ( z) β y σ y div(κ). Te pressure is eliminated from te orizontal momentum equations using (3.3) and (3.4). Te details will be given for te x momentum equation and ten te result will be stated for te y momentum equation. (3.2) (3.3) (3.4) 4. Reduction of te orizontal momentum equation Te x component of te momentum equations (2.4) is Du + 1 p ρ x = 2(Ω 2w Ω 3 v) Ω 2 (z + d 3 ) + Ω 3 (y + d 2 ) Ω 1 Ω (x + d) + (x + d 1 ) Ω 2 Qe 1 q gqe 1 e 3. Replace te second term on te left-and side by te expression for ρ 1 p x in (3.3), (4.1)

8 8 H. Alemi Ardakani and T. J. Bridges Du + Du z + Dw x = 2Ω 2 W + 2Ω 3 V 2 Ω 2 ( z) 2Ω 1 V x + 2Ω 2 U x + 2Ω 2 w 2Ω 3 v +(Ω Ω 2 2)( + d 3 ) x + β x ( z) + β x + σ x div(κ) 2(Ω 2 w + Ω 3 v) Ω 2 (z + d 3 ) + Ω 3 (y + d 2 ) Ω 1 Ω (x + d) + (x + d 1 ) Ω 2 Qe 1 q gqe 1 e 3. Tere are convenient cancellations: principally Du, 2 Ωz and te interior Coriolis terms all cancel out. Cancelling and using β x = Ω 2 Ω 1 Ω 3, and te kinematic condition W = t + U x + V y gives Du + Dw x + 2Ω 2 ( t + U x + V y ) 2Ω 3 V + 2Ω 1 V x 2Ω 2 U x (Ω Ω 2 2)( + d 3 ) x (Ω Ω 2 3)(x + d 1 ) + Ω 1 Ω 2 (y + d 2 ) + Ω 1 Ω 3 ( + d 3 ) + Ω 2 ( + d 3 ) Ω 3 (y + d 2 ) + Qe 1 q + gqe 1 e 3 β x σ x div(κ) = 0. Now use te fact tat Du can be expressed purely in terms of surface variables since, U t + UU x + V U y = Du Substition into (4.2) reduces te x momentum equation to an equation purely in terms of surface variables ( U t + UU x + V U y + a 11 + Dw ) x + a 12 y = b 1 + σ x div(κ). (4.3) Te coefficients in tis equation are and a 11 (x, y, t) = 2Ω 1 V (Ω Ω 2 2)( + d 3 ) β = 2Ω 1 V + Qe 3 q + gqe 3 e 3 (Ω Ω 2 2)( + d 3 ) ( Ω 2 Ω 1 Ω 3 )(x + d 1 ) + ( Ω 1 + Ω 3 Ω 2 )(y + d 2 ),. (4.2) (4.4) a 12 = 2Ω 2 V, (4.5) b 1 (x, y, t) = 2Ω 2 t + 2Ω 3 V Qe 1 q gqe 1 e 3 + (Ω Ω 2 3)(x + d 1 ) +( Ω 3 Ω 1 Ω 2 )(y + d 2 ) ( Ω 2 + Ω 1 Ω 3 )( + d 3 ). Te use of te unit vectors e 1, e 2 and e 3 is just to compactify notation. Te terms wit unit vectors are interpreted as follows Qe 3 e 3 = Q 33, were Q ij is te (i, j) t entry of te matrix representation of Q and Qe 3 q = Q 13 q 1 + Q 23 q 2 + Q 33 q 3, wit similar expressions for te oter suc terms. A similar argument (see Appendix A.3) leads to te surface y momentum equation ( V t + UV x + V V y + a 21 x + a 22 + Dw ) y = b 2 + σ y div(κ). (4.7) (4.6) wit a 21 = 2Ω 1 U, (4.8)

9 and and Sallow water slosing in 3D 9 a 22 = 2Ω 2 U (Ω Ω 2 2)( + d 3 ) + Qe 3 q + gqe 3 e 3 +( Ω 1 + Ω 2 Ω 3 )(y + d 2 ) ( Ω 2 Ω 1 Ω 3 )(x + d 1 ). b 2 = 2Ω 1 t 2Ω 3 U Qe 2 q gqe 2 e 3 + (Ω Ω 2 3)(y + d 2 ) Te terms a 11 and a 22 are related by ( Ω 3 + Ω 1 Ω 2 )(x + d 1 ) + ( Ω 1 Ω 2 Ω 3 )( + d 3 ). a 11 2Ω 1 V = a Ω 2 U. (4.9) (4.10) Te surface SWEs (4.3) and (4.7) are exact. Moreover te assumption of finite dept as not been used yet and so tey are also valid in infinite dept. In order to reduce tem to a closed system, te only term tat requires modelling is te Lagrangian vertical acceleration at te surface, Dw. 5. Penney-Price-Taylor teory for te igest standing wave Before proceeding to reduce te surface equations to a closed set of sallow water equations a key property of te exact equations is igligted. One of te simplest forms of slosing waves is te pure standing wave. It is periodic in bot space and time. Penney & Price (1952) argue tat te igest standing wave sould occur wen te Lagrangian vertical acceleration at te crest is equal to g. Tey consider standing waves in infinite dept only, but it will be clear from te discussion below tat teir argument is also valid in finite dept. Teir argument in te absence of surface tension is tat te pressure just inside te liquid near te surface must be positive or zero and consequently at te surface p z 0 wic is equivalent to g + Dw 0. (5.1) Wen tis condition is violated te standing wave sould cease to exist. Taking into account tat Dw = w t at a crest, tis is equation (67) in Penney & Price (1952). Using tis teory tey deduced tat te crest angle of te igest wave must be 90 o, in contrast to te 120 o angle of travelling waves. Taylor (1953) was surprised by tis argument and tested it by constructing an experiment. He was mainly interesting in te crest angle. His experiments convincingly confirmed te conjecture of Penney & Price (1952). A teoretical justification of tis teory can be deduced from te surface momentum equations. For te case of 2D waves, tis argument as been presented in Appendix F of Alemi Ardakani & Bridges (2009g). Remarkably tis argument carries over to 3D waves. Neglecting surface tension, and assuming te vessel to be stationary, te surface momentum equations (4.3) and (4.7) reduce to ( U t + UU x + V U y + g + Dw ) x = 0 ( V t + UV x + V V y + g + Dw ) y = 0. Wen g + Dw = 0, tese equations reduce to U t + UU x + V U y = 0 V t + UV x + V V y = 0. Tese equations are closed and indeed it is sown by Pomeau et al. (2008a) tat tey ave an exact similarity solution. Moreover tis similarity solution gives a form of wave breaking, wic as in turn been confirmed by numerical experiments in Pomeau et al. (2008b). Te teoretical argument in Pomeau et al. (2008a) is by analogy wit te sallow water equations but it is sown to be precise in Bridges (2009). Tis teory indicates tat any 3D standing waves will be susceptible to some form of

10 10 H. Alemi Ardakani and T. J. Bridges breaking wen te condition (5.1) is violated. Indeed potograps of te experiments of Taylor (1953) sow a form of crest instability near te igest standing wave in te 2D case, and is experiments also sow a tendency to 3D near te igest wave. See also Figure 8(c) in Kobine (2008) wic sows 3D standing waves reacing a maximum. Adding in surface tension and a rotating frame will add new features to te teory. Surface tension will likely provide a smooting effect, but rotation will likely lead to a more complicated scenario for breaking. In tis paper we are interested in sallow water slosing. In tis case it is natural to assume tat te Lagrangian vertical accelerations at te surface are small, and so we will be working predominantly in te region were te condition (5.1) is strongly satisfied. 6. Conservation of mass Te vertical average of te orizontal velocity is (u(x, y, t), v(x, y, t)) defined by Differentiating u := 1 0 u(x, y, z, t) dz and v := 1 0 v(x, y, z, t) dz. (6.1) t + (u) x + (v) y = t + x u + 0 u x dz + y v + 0 v y dz = t + U x + V y + 0 (u x + v y + w z ) dz 0 w z dz = t + U x + V y W + w = 0, using u x + v y + w z = 0 = 0, te bottom boundary condition and te kinematic free surface boundary condition. Hence, if (u, v) are used for te orizontal velocity field ten te equation in te SWEs in te form z=0 t + (u) x + (v) y = 0, (6.2) is exact. However we are interested in an equation based on te surface orizontal velocity field. Te surface and average velocities are related by U(x, y, t) u(x, y, t) = 1 V (x, y, t) v(x, y, t) = zu z dz, zv z dz. Use tese identities to formulate te mass equation in terms of te surface velocity field. Differentiating (6.3) and using mass conservation, Replace W by te kinematic condition, x [(U u)] + y [(V v)] = W + U x + V y. ((U u)) x + ((V v)) y = W + (U x + V y ) = t + (U) x + (V ) y. Te error in using te surface velocity field in te equation can be caracterized two ways: (6.3) W + (U x + V y ) 0 or ((U u)) x + ((V v)) y 0. (6.4)

11 7. SWEs for 3D slosing in a rotating vessel To summarize, te candidate SWEs for (, U, V ) are Sallow water slosing in 3D 11 t + (U) x + (V ) y = W + U x + V y ( ) U t + UU x + V U y + a 11 + Dw x + a 12 y = b 1 + σ x div(κ) ( ) V t + UV x + V V y + a 21 x + a 22 + Dw y = b 2 + σ y div(κ). (7.1) Te equation for W can be added W t + UW x + V W y = Dw. (7.2) Te system (7.1) wit or witout (7.2) is not closed. If Dw is specified, ten te system of four equations (7.1)-(7.2) for (, U, V, W ) is closed. Tis system of four equations can be furter reduced to a system of 3 equations wit an additional assumption on te surface vertical velocity. Hencefort it is assumed tat te vertical velocity at te free surface satisfies W + U x + V y << 1, (SWE-1) and te Lagrangian vertical acceleration at te free surface satisfies Dw << a 11 and Dw << a 22. (SWE-2) Te assumption (SWE-1) as an alternative caracterization as sown in (6.4). For small vessel motion, te second assumption (SWE-2) is equivalent to assuming tat te Lagrangian vertical accelerations are small compared wit te gravitational acceleration, equivalent to (5.1). Under te assumptions (SWE-1) and (SWE-2) and under te additional assumption tat σ = 0 (neglect of surface tension), te SWEs are yperbolic. Wen σ 0 tey are dispersive, for in tat case div(κ) = κ 1 x + κ 2 y = xx + yy +, were te dots correspond to nonlinear terms in and its derivatives. Hence te σ terms in te rigt-and side of (U, V ) equations in (7.1) ave te form x div(κ) = xxx + yyx +, y div(κ) = xxy + yyy +. At te linear level, tese terms add dispersion to te SWEs. Tey will also require additional boundary conditions at te walls (cf. Billingam 2002; Kidambi & Sankar 2004), as a contact angle effect appears at te vessel walls. In tis paper we will be primarily concerned wit long waves and so will encefort neglect surface tension: σ = 0. (SWE-3) 8. Prescribing te rigid-body motion of te vessel Te fluid vessel is a rigid body free to undergo any motion in 3 dimensional space. Every rigid body motion in R 3 is uniquely determined by (q(t), Q(t)) were q(t) is te 3 component translation vector and Q(t) is an ortogonal matrix wit unit determinant (cf. Capter 7 of O Reilly 2008). In terms of te body coordinates, te translations are surge, sway and eave, and te rotations are labelled roll, pitc and yaw as illustrated in Figure 2. Translations are straigtforward to prescribe and need no special attention oter tan to be careful

12 12 H. Alemi Ardakani and T. J. Bridges z eave yaw sway pitc y roll x surge Figure 2. Diagram sowing conventions for roll, pitc, yaw, surge, sway and eave. about weter te spatial or body representation is used. In tis paper q(t) is te translation of te body relative to te spatial frame. If te translation is specified from onboard te vessel tat is, specifying te surge, sway and eave directly ten te accelerations are related by q surge 1 q sway 2 q eave 3 Q 11 Q 21 Q 31 q 1 = Q 12 Q 22 Q 32 q 2. Q 13 Q 23 Q 33 q 3 Te natural approac in te context of expermiments is to specify te absolute translations along wit te rotations. Altoug we are not aware of any experiments wic combine bot. Te paper of Disimile et al. (2009) indicates tat teir experimental facility for slosing as te capability to produce all 6 degrees of freedom in te forcing, but only planar motions are reported so far. On te oter and specification of te rotations requires some care. Te set of ortogonal matrices is igly nonlinear and in R 3 te rotation matrices are no longer commutative in general. Te simplest way to specify a rotation is to use Euler angles. However, even ere one must be careful because Euler angle representations are inerently singular, and teir are subtleties in te deduction of te appropriate angular velocity. Te construction of Q(t) can also be approaced directly using numerical integration. Te rotation matrix satisfies te differentiation equation Q = Q Ω, Q(0) = I, (8.1) were Ω is te matrix representation of te body angular velocity (2.1). Tis approac is most effective if te angular velocity is given. One setting were te body angular velocity is available is in strapdown inertial navigation systems (e.g. Capter 11 of Titterton & Weston 2004). Te navigation system outputs te body angular velocity and ten (8.1) is solved numerically for Q(t) (called te attitude matrix in navigation literature). Anoter efficient approac to constructing rotation matrices is te use of quaternions. Quaternions are now widely used in computer grapics algoritms (Hanson 2006) and in molecular dynamics (Evans 1977; Rapaport 1985). Evans (1977) points out tat quaternion [computer] programmes seem to run ten times faster tan corresponding [computer] programmes employing Euler angles. However in our case te computing time for te rotations is very small compared to te computing time for te fluid motion, so te simpler approac of computing Q directly by integrating (8.1) is adapted. Te differential equation (8.1) can be integrated numerically very efficiently, altoug te coice of numerical integrator is very important as it is essential to maintain ortogonality to macine accuracy.

13 Sallow water slosing in 3D 13 z Z θ,pitc y Y X, x φ, roll Figure 3. Scematic of te roll-pitc motion in terms of Euler angles φ and θ. An efficient second-order algoritm for (8.1) is te implicit midpoint rule (Leimkuler & Reic 2004) wit discretization Q n+1 Q n ( Q n+1 + Q n ) ( ) = Ω t n+ 1 2, t 2 were Ω(t) is treated as given; rearranging Setting one time step is represented by Q n+1 = Q n t ( Q n+1 + Q n) Ω S n+ 1 ( ) 2 = 1 2 t Ω t n+ 1 2, Q n+1 = Q n (I + S n+ 1 2 ( ) t n ) ( I S n+ 1 2 ) 1. (8.2) Since S n is skew-symmetric, te term (I+S n )(I S n ) 1 is ortogonal. Hence ortogonality is preserved to macine accuracy at eac time step. Te implicit midpoint rule is a special case of te Gauss-Legendre Runge-Kutta (GLRK) metods wic ave been sown to preserve ortogonality to macine accuracy, and tey can be constructed to any order of accuracy (Leimkuler & Reic 2004). Tere are oter effective and efficient metods for integrating te equation (8.1) and some recent developments are reviewed in Romero (2008) Euler angles roll-pitc rotation Roll-pitc motion is te simplest non-commutative rotation tat is of interest in sip dynamics. Te roll-pitc excitation as been used by a number of autors for forced slosing (e.g. Pantazopoulos 1988, 1987; Faltinsen et al. 2006b; Huang & Hsiung 1996, 1997; Huang 1995). In tis subsection te properties of te Euler angle representation of roll-pitc are recorded. Te roll-pitc rotation consists of a counterclockwise roll rotation about te x axis wit angle φ, followed by a counterclockwise pitc rotation about te current y axis, as illustrated scematically in Figure 3. After converting bot rotations to te same basis te rotation matrix takes te form cos θ 0 sin θ Q = sin φ sin θ cos φ sin φ cos θ. (8.3) cos φ sin θ sin φ cos φ cos θ

14 14 H. Alemi Ardakani and T. J. Bridges From tis expression te body representation of te angular velocity is easily deduced using Q T Q = Ω and so φ cos θ Ω = θ φ sin θ. (8.4) Te spatial angular velocity is obtained by multiplication of Ω by Q. A derivation of te roll-pitc angular velocity from first principles is given in Alemi Ardakani & Bridges (2009d). If θ and φ are considered as small, ten te approximate body angular velocity is Ω ( φ, θ, 0 ), (8.5) obtained by neglecting quadratic and iger-order terms in (8.4). Tis simplified version of te angular velocity as been used by Pantazopoulos (1988, 1987), Falzarano et al. (2002) and Faltinsen et al. (2006b). A significant problem wit tis simplification is tat te rotation as been qualitatively canged: te body and spatial represenations are equal and so it beaves like a planar rotation. Secondly, in a numerical context tis simplification is not necessary Yaw-pitc-roll rotation and Euler angles Te yaw-pitc-roll rotation is one of te most widely used Euler angle sequences (see of O Reilly (2008) were tey are called te Euler angle sequence). It was first used in te context of slosing by Huang (1995) and Huang & Hsiung (1996, 1997). Te Euler angle sequence starts wit a yaw rotation about te z axis wit angle ψ, followed by a pitc rotation about te new y axis denoted by θ, followed by a roll rotation about te new x axis denoted by φ. Te composite rotation is cos θ cos ψ sin φ sin θ cos ψ cos φ sin ψ cos φ sin θ cos ψ + sin φ sin ψ Q = cos θ sin ψ sin φ sin θ sin ψ + cos φ cos ψ cos φ sin θ sin ψ sin φ cos ψ. (8.6) sin θ sin φ cos θ cos φ cos θ Te body angular velocity is computed to be φ ψ sin θ Ω = ψ cos θ sin φ + θ cos φ. (8.7) ψ cos θ cos φ θ sin φ Full details of te Euler angles and te derivation of te angular velocity are given in Alemi Ardakani & Bridges (2009c). In matrix form te angular velocity is related to te Euler angles by wit Ω = B 1 Θ, 1 sin φ tan θ cos φ tan θ φ B = 0 cos φ sin φ and Θ := θ. 0 sin φ sec θ cos φ sec θ ψ Tis is te form of te angular velocity used in Huang (1995) and Huang & Hsiung (1997). Te singularity of tis Euler angle representation arises due to te non-invertibility of B: det(b) = sec θ, and so to avoid te singularity te restriction 1 2 π < θ < 1 2π is required. In te context of sip dynamics tis is not a severe restriction. If all tree angles are small ten B is approximately te identity and φ Ω θ. ψ

15 Sallow water slosing in 3D 15 Tis approximation is appealing, since ten Ω = Θ and Ω = Θ, wit Θ = (φ, θ, ψ). But te spatial and body representations are equal, and so te approximate rotation is qualitatively different from te exact rotation. In te numerical setting an approximation is not needed and te exact expression is used Harmonic forcing Harmonic motion can be specified by expressing te Euler angles in terms of armonic motion. For example in te case of roll-pitc wit te same frequency and pase, but different amplitudes, φ(t) = ε 1 cos ωt and θ(t) = ε 2 cos ωt. In coosing te forcing frequency it is not te value of te frequency tat is important but its value relative to te natural frequency. In te limit of sallow water, te natural frequencies of te fluid are ω mn = π ( ) m 2 1/2 g 0 L 2 + n2 1 L 2. (8.8) 2 9. Review of previous derivations of rotating SWEs for slosing Two derivations of te SWEs for fluid in a vessel tat is undergoing a general rigid-body motion in tree dimensions first appeared in te literature at about te same time, given independently by Dillingam & Falzarano (1986) and Pantazopoulos (1987). Bot derivations follow te same strategy. Teir respective derivations are an extension of te formulation for two-dimensional sallow water flow in a rotating frame in Dillingam (1981). Tis review of teir derivation serves two purposes. It sows ow te coice of representation of te angular velocity is very important, and secondly, it sows ow te new SWEs in 7 compare wit previous work. Hereafter tese equations (introduced below in equation (9.2)) will be called te DFP SWEs. Te DFP derivation starts wit te classical SWEs u t + uu x + vu y + g x = 0 v t + uv x + vv y + g y = 0 t + (u) x + (v) y = 0. Ten replace g wit te vertical acceleration of te moving frame, and add te orizontal accelerations to te rigt and side of te two momentum equations, leading to (9.1) wit u t + uu x + vu y + a (z) x = f 1 v t + uv x + vv y + a (z) y = f 2 t + (u) x + (v) y = 0, f 1 = n 1 cos θ n 2 sin φ sin θ + n 3 sin θ cos φ 2ω 1 ω 2 x sin φ sin θ cos θ ω 1 ω 2 y cos φ cos θ ω 1 ω 2 z d sin φ(sin 2 θ cos 2 θ) + ω1x 2 sin 2 θ ω1z 2 d sin θ cos θ + ω2x(1 2 sin 2 θ sin 2 φ) ω2y 2 sin φ sin θ cos φ +ω2z 2 d sin θ sin 2 φ cos θ + 2ω 1 v sin θ 2ω 2 v sin φ cos θ + ω 1 y sin θ + ω 2 y sin φ cos θ ω 2 z d cos φ + g sin θ cos φ, f 2 = n 2 cos φ n 3 sin φ + ω1y 2 ω2x 2 sin φ sin θ cos φ + ω2y 2 sin 2 φ +ω2z 2 d sin φ cos φ cos θ ω 1 ω 2 x cos φ cos θ ω 1 ω 2 z d sin θ cos φ 2ω 1 u sin θ + 2ω 2 u sin φ cos θ ω 1 x sin θ + ω 1 z d cos θ + ω 2 x sin φ cos θ + ω 2 z d sin θ sin φ g sin φ, (9.2) (9.3) (9.4)

16 16 H. Alemi Ardakani and T. J. Bridges and te vertical acceleration is a (z) = n 1 sin θ n 2 sin φ cos θ + n 3 cos φ cos θ + ω 1 ω 2 x sin φ(sin 2 θ cos 2 θ) +ω 1 ω 2 y sin θ cos φ 2ω 1 ω 2 z d sin φ sin θ cos θ +ω1x 2 sin θ cos θ ω1z 2 d cos 2 θ ω2x 2 sin θ sin 2 φ cos θ ω2y 2 sin φ cos φ cos θ ω2z 2 d (1 sin 2 φ cos 2 θ) + 2ω 1 v cos θ +2ω 2 v sin φ sin θ 2ω 2 u cos φ + ω 1 y cos θ ω 2 x cos φ + ω 2 y sin φ sin θ + g cos φ cos θ. (9.5) Te notation follows te original sources, and several typos ave been corrected. n = (n 1, n 2, n 3 ) in tese equations is te same as q in 2, and ω = (ω 1, ω 2, ω 3 ) is te spatial representation of te angular velocity. Te oter terms will be defined sortly. At first glance tis formulation does not look anyting like te rotating SWEs proposed in 7. Tis discrepancy concerned us greatly, and we undertook a careful investigation of te assumptions and properties of te DFP equations. Te details of tat investigation, including correction of te various typos and omissions, is recorded in te report of Alemi Ardakani & Bridges (2009d). Here we will igligt te main features of te derivation leading to (9.2). Let X = (X, Y, Z) be coordinates for te fixed spatial frame and let x = (x, y, z) be coordinates for te body frame. Ten DFP use te following relationsip between te fixed frame and te moving frame X = Q(x + d) + n, wit n identified wit q in 2. Tey impose te following assumptions on x and d x x = y, 0 and 0 d = 0. z d (DFP-1) (DFP-2) Te implication of assumption (DFP-1) is tat tey are restricting te 3D equations to te plane z = 0. Te second assumption (DFP-2) is just a coice of axis of rotation and could be generalized witin te DFP derivation. Te only forcing considered is roll-pitc of te form in 8.1. Te rotation matrix is restricted to cos θ 0 sin θ Q = sin φ sin θ cos φ sin φ cos θ. (DFP-3) cos φ sin θ sin φ cos φ cos θ Tis matrix is given explicitly in equation (6) on page 29 of Pantazopoulos (1987). A key assumption in tese equations is te use of spatial angular velocity rater tan body angular velocity and te assumption ω 3 = 0. (DFP-4) Tis assumption is a curiosity. It is certainly not necessary in a numerical context, and it is inconsistent since te tird component of te body angular velocity is not zero. Te body angular velocity and spatial angular velocity are related by φ cos θ Ω = Q T ω = θ φ sin θ.

17 Assuming θ and φ small gives Sallow water slosing in 3D 17 φ φ ω θ and Ω θ. φ θ θ φ If we furter neglect te quadratic terms φ θ and θ φ ten te angular velocities reduce to φ ω = Ω = θ. 0 Tis is te approximation used explicitly by Falzarano et al. (2002) and is implicitly used in Pantazopoulos (1988, 1987). Formalizing tis assumption φ 0 and θ 0. (DFP-5) However, te development of te DFP SWEs does not invoke assumption (DFP-5) until after te equations are derived. Wit just assumption (DFP-4) te spatial and body accelerations are related by and Ω 1 = ω 1 cos θ + ω 2 sin φ sin θ Ω 2 = ω 2 cos φ Ω 3 = ω 1 sin θ ω 2 sin φ cos θ. Using assumption (DFP-3) te translation and gravity terms can be expressed using Q, n Qe 1 = n 1 cos θ + n 2 sin φ sin θ n 3 cos φ sin θ, ge 3 Qe 1 = g cos φ sin θ. Using tese two expressions, (9.6), and some calculation we find tat f 1 simplifies to (9.6) f 1 = n Qe 1 + (Ω Ω 2 3)x Ω 1 Ω 2 y Ω 1 Ω 3 z d +2Ω 3 v Ω 2 z d + Ω 3 y ge 3 Qe 1. (9.7) A similar construction sows tat f 2 = n Qe 2 + (Ω Ω 2 3)y Ω 1 Ω 2 x Ω 2 Ω 3 z d 2Ω 3 u + Ω 1 z d Ω 3 x ge 3 Qe 2. (9.8) Te simplification in te expressions (9.7) and (9.8) over te original expressions (9.3) and (9.4) is remarkable. Te simplification is due first to te use of te body angular velocity and secondly to te explicit use of te rotation operator. Tese formulae are verified by substitution and te details are given in Alemi Ardakani & Bridges (2009d). Te vertical acceleration term can also be simplified to a (z) = n Qe 3 (Ω Ω 2 2)z d + Ω 1 Ω 3 x + Ω 2 Ω 3 y +2Ω 1 v 2Ω 2 u + Ω 1 y Ω 2 x + ge 3 Qe 3. (9.9) Now tat te DFP SWEs look a little more like te surface SWEs in 7 we can compare te two systems. Te velocity field (U, V ) is not te same as te velocity field in te DFP SWEs. DFP do not specify exactly wic orizontal velocity tey use but teir equation becomes exact if te average orizonatal velocity is used. In any case, assume for purposes of comparison tat (U, V ) (u, v), and

18 18 H. Alemi Ardakani and T. J. Bridges ten te coefficients in te two systems can be compared. a 11 = a (z) + 2Ω 2 U (Ω Ω 2 2) ( Ω 2 Ω 1 Ω 3 )d 1 + ( Ω 1 + Ω 2 Ω 3 )d 2 a 12 = a DFP Ω 2 V a 21 = a DFP 21 2Ω 1 U a 22 = a (z) 2Ω 1 V (Ω Ω 2 2) ( Ω 2 Ω 1 Ω 3 )d 1 + ( Ω 1 + Ω 2 Ω 3 )d 2. In te DFP SWEs a DFP 12 and a DFP 21 are identically zero but teir symbols are included for comparison. Te rigt-and side coefficient comparison is b 1 = f 1 2Ω 2 t + (Ω Ω 2 3)d 1 + ( Ω 3 Ω 1 Ω 2 )d 2 ( Ω 2 + Ω 1 Ω 3 ) b 2 = f 2 + 2Ω 1 t + (Ω Ω 2 3)d 2 ( Ω 3 + Ω 1 Ω 2 )d 1 + ( Ω 1 Ω 2 Ω 3 ) Te d 1 and d 2 error terms are not so important since tey could be included in te DFP formulation so tey can be discounted. Te discrepancy between te two formulations is still quite significant wen te rotation field is present. Tis discrepancy is due to te fact tat te DFP SWEs ave more assumptions tan te surface SWEs Review of te HH SWEs Anoter strategy for deriving te SWEs for fluid in a vessel tat is undergoing a general rigid-body motion in tree dimensions as been proposed by Huang (1995) and Huang & Hsiung (1996, 1997). Teir derivation is more precise, and starts wit te full 3D equations. A detailed report on teir derivation is given in Alemi Ardakani & Bridges (2009e). Here a sketc of teir derivation is given including a comparison wit te DFP SWEs and te surface equations in 7. Tey explicitly take u(x, y, z, t) and v(x, y, z, t) to be independent of z but implicitly tey are using te average orizontal velocity field (u, v). Tey neglect te vertical acceleration in general (not just at te free surface) and integrate te vertical pressure gradient, differentiate and ten substitute into te orizontal momentum equations. However due to tese assumptions extraneous terms appear wic require additional assumptions: 2Ω 2 w 0, 2Ω 1 w 0, (Ω 1 v x Ω 2 u x ) 0, (Ω 1 v y Ω 2 u y ) 0. Te rotation matrix Q is restricted to te yaw-pitc-roll rotation discussed in 8.2, and tey use te body angular velocity representation. Teir derivation leads to SWEs of te form u t + uu x + vu y + a HH 11 x = b HH 1, v t + uv x + vv y + a HH 22 y = b HH 2, t + (u) x + (v) y = 0, (9.10) were te (u, v) velocity field ere sould be interpreted as te average velocity field (u, v). To compare tese HH SWEs wit new surface equations, assume tat te (U, V ) velocity field in te surface equations is equivalent to te velocity field in te HH SWEs, and compare coefficients a HH 11 = a 11 2Ω 2 u, a HH 22 = a Ω 1 v, a HH 12 = a 12 2Ω 2 v, a HH 21 = a Ω 1 u, b HH 1 = b 1 + 2Ω 2 t + Ω 2, b HH 2 = b 2 2Ω 1 t Ω 1. (9.11) Te agreement between te HH SWEs and te surface SWEs is muc better tan te case of te DFP SWEs but tere are still important differences wen Ω 1 and Ω 2 are important. One case were

19 Sallow water slosing in 3D 19 te surface equations and te HH SWEs agree exactly (assuming equivalence of te velocity fields) is wen te forcing is pure yaw motion, and numerical experiments on tis case are discussed in Numerical algoritm for sallow-water slosing Te numerical metod we propose for simulation of sallow water slosing in a veicle undergoing rigid body motion is an extension of te numerical sceme in Alemi Ardakani & Bridges (2009g). It is a finite difference sceme, using centered differencing in space. It is fully implicit and as a blocktridiagonal structure. Te basic formulation of te algoritm was first proposed by Leendertse (1967) and refined by Abbott and Ionescu and is widely used in computational ydraulics (cf. Abbott 1979). Te only new features in te algoritm are extension to include a fully 3D rotation and translation field, and exact implementation of boundary conditions. Te fact tat te sceme is implicit makes te introduction of rotations straigtforward. Some explicit scemes are unstable in te presence of rotation. Te one-dimensional version of te algoritm was used in Alemi Ardakani & Bridges (2009g) and te two-dimensional version is just a concatentation of tis sceme: te time step is split into two steps and an alternating direction implicit algoritm is used: implicit in x direction and explicit in te y direction in te first alf step and explicit in x direction and implicit in te y direction in te second alf step. One of te nice properties of te sceme is tat te boundary conditions at te walls are implemented exactly, even at te intermediate time steps. Te sceme as numerical dissipation, but te form of te dissipation is similar to te action of viscosity. Te truncation error is of te form of te eat equation and so is strongly wavenumber dependent. Moreover te numerical dissipation follows closely te ydraulic structure of te equations. See te tecnical report of Alemi Ardakani & Bridges (2009f ) for an analysis of te form of te numerical dissipation. Te numerical dissipation is elpful for eliminating transients and spurious ig-wavenumber oscillation in te formation of travelling ydraulic jumps. In contrast, Pantazopoulos (1987) and Pantazopoulos (1988) use Glimm s metod. Glimm s metod is very effective for treating a large number of travelling ydraulic jumps, but te solutions are discontinuous, and te sceme as problems wit mass conservation. Huang & Hsiung (1996, 1997) use flux-vector splitting. Tis metod involves computing eigenvalues of te Jacobian matrices, and is effective for tracking multi-directional caracteristics. It appears to be very effective and accurate but is more complicated to implement tan te sceme proposed ere. Setting up te equations for tis sceme is straigtforward and follows te one-dimensional construction in Alemi Ardakani & Bridges (2009g), but te details are lengty. Hence we ave given a detailed construction of te algoritm in an internal report (Alemi Ardakani & Bridges 2009f ) and in tis section just te main features are igligted. Rewrite te governing equations in a form suitable for te first alf-step of te sceme, t + U x + U x + V y + V y = 0 U t + U U x + V U y + 2Ω 2 V y + 2Ω 2 t + [ α(x, y, t) + 2Ω 1 V ( Ω Ω 2 2) ] x = 2Ω 3 V ( Ω2 + Ω 1 Ω 3 ) + β(x, y, t) (10.1) V t + U V x + V V y 2Ω 1 U x 2Ω 1 t + [ α(x, y, t) 2Ω 2 U ( Ω Ω 2 2) ] y = 2Ω 3 U + ( Ω1 Ω 2 Ω 3 ) + β(x, y, t),

20 20 H. Alemi Ardakani and T. J. Bridges y j=jj, y=l 2 j=1 x i=1 Figure 4. A scematic of te grid layout. i=ii, x=l 1 were α, β and β are te terms tat are independent of, U and V, α(x, y, t) = ( ( ) Ω Ω2) 2 d3 + Ω1 + Ω 2 Ω 3 (y + d 2 ) + (Ω 1 Ω 3 Ω ) 2 (x + d 1 ) +Qe 3 q + gqe 3 e 3 ( ) ( ) β(x, y, t) = Ω2 + Ω 1 Ω 3 d 3 + Ω3 Ω 1 Ω 2 (y + d 2 ) + ( Ω Ω3) 2 (x + d1 ) β(x, y, t) = Qe 1 q gqe 1 e 3 ( ) ( ) Ω1 Ω 2 Ω 3 d 3 Ω3 + Ω 1 Ω 2 (x + d 1 ) + ( Ω Ω3) 2 (y + d2 ) Qe 2 q gqe 2 e 3. (10.2) Te terms wit superscript are nonlinear terms tat are treated implicitly. In tis first alf step only x derivatives are implicit and y derivatives are explicit. Te nonlinearity is addressed using iteration. Te x interval 0 x L 1 is split into II 1 intervals of lengt x = x i := (i 1) x, i = 1,..., II, and te y interval 0 y L 2 is split into JJ 1 intervals of lengt y = and y j := (j 1) y, j = 1,..., JJ, L1 II 1 L2 JJ 1 and so and so n i,j := (x i, y j, t n ), U n i,j := U(x i, y j, t n ) and V n i,j := V (x i, y j, t n ), were t n = n t wit t te fixed time step. A scematic of te grid is in Figure 4. Te discretization of te mass equation is n+ 1 2 i,j n i,j 1 2 t + i,j U n+ 1 2 i+1,j U n+ 1 2 i 1,j + U n+ 1 2 i,j 2 x i+1,j 1 n+ 2 i 1,j 2 x + n Vi,j+1 n V i,j 1 n i,j + V n n i,j+1 n i,j 1 i,j = 0. 2 y 2 y (10.3)

21 Te discretizations of te equations for U, V are Sallow water slosing in 3D 21 U n+ 1 2 i,j U n i,j U 2 t i,j U n+ 2 1 i+1,j U n+ 1 2 i 1,j 2 x + Vi,j n U n i,j+1 U n i,j 1 2 y + 2Ω n Vi,j n n i,j+1 n i,j 1 2 y [ ( ( ) α n+ 1 2 i,j + 2Ω n Vi,j Ω n+ 1 2 ( Ω n ( ) = 2Ω n V n+ 1 2 i,j 1 Ωn Ω n Ω n n+ 1 2 i,j 2Ω n ) 2 ) ] n i+1,j n+ 2 i 1,j i,j 2 x n+ 1 2 i,j n i,j 1 + β n t i,j (10.4) were V n+ 1 2 i,j + By setting V n i,j 1 + U 2 t i,j V n+ 1 2 i+1,j V n+ 1 2 i 1,j 2 x + Vi,j n V n i,j+1 V n i,j 1 2 y 2Ω n Ui,j n+ 1 2 ] [ ( ( α n+ 1 2 i,j 2Ω n Ui,j n Ω n ) 2 + ( Ω n ) 2 ) n i,j ( ) = 2Ω n U n+ 1 2 i,j + 1 Ωn+ 2 1 Ω n Ω n n+ 1 2 i,j + 2Ω n n i,j+1 n i,j 1 2 y 1 i+1,j n+ 2 i 1,j 2 x n+ 1 2 i,j n i,j 1 + β n t α n i,j := α(x i, y j, t n ), βn i,j := β(x i, y j, t n ) and βn i,j := β(x i, y j, t n ). n i,j z n i,j = U i,j n, Vi,j n equations (10.3)-(10.4) can be written in block tridiagonal form i,j, A n+ 1 2 i,j z n+ 1 2 i 1,j + Bn+ 1 2 z n+ 1 2 i,j + A n+ 1 2 i,j z n+ 1 2 i+1,j = C n+ 1 2 i,j z n i,j 1 + Dn+ 1 2 z n i,j C n+ 1 2 i,j z n i,j+1 + βn+ 1 2 i,j. (10.5) Te left subscript is an indication tat te matrix depends on and/or U. Expressions for te matrices are given in (Alemi Ardakani & Bridges (2009f )). For fixed j = 2,..., JJ 1 te equations (10.5) are applied for i = 2,..., II 1. To complete te tridiagonal system equations are needed (for eac fixed j) at i = 1 and i = II Te equations at i = 1 and i = II for j = 2,..., JJ 1 Te equations at i = 1 and i = II are obtained from te boundary conditions at x = 0 and x = L 1. Te only boundary condition at x = 0 is U(0, y, t) = 0. Te discrete version of tis is U n 1,j = 0 and U n 0,j + U n 2,j 2 To obtain a boundary condition for, use te mass equation at x = 0 wit discretization n ,j + t 2 x 1,j U n ,j = n 1,j t 4 y n 1,j = 0, for eac j, and for all n N. (10.6) t + U x + V y + V y = 0, ( ) V n 1,j+1 V1,j 1 n t 4 y V ( ) 1,j n n 1,j+1 n 1,j 1. (10.7) To obtain a boundary condition for V, use te y momentum equation at x = 0 V t + V V y 2Ω 1 t + [ α(x, y, t) ( Ω Ω 2 ] ( ) 2) y = Ω1 Ω 2 Ω 3 + β(x, y, t),

22 22 H. Alemi Ardakani and T. J. Bridges wit discretization were V n ,j [ ( 2Ω n t 1 Ωn+ 2 1 Ω n Ω n )] n ,j = V1,j n t 4 y V ( ) 1,j n V n 1,j+1 V1,j 1 n ( n 1,j+1 1,j 1) n t 4 y α n ,j α n i,j = α n i,j ((Ω n 1 ) 2 + (Ω n 2 ) 2) n 1 2 i,j. Combining equations (10.6), (10.7) and (10.8) gives te equation for i = 1 2Ω n n 1,j t β n ,j, (10.8) E n+ 1 2 z n ,j + F 1,j z n ,j = G n ,j z n 1,j 1 + Hn+ 1 2 z n 1,j Gn ,j z n 1,j β n ,j. (10.9) Expressions for te matrices are given in Alemi Ardakani & Bridges (2009f ). A similar strategy is used to construct te discrete equations at x = L Summary of te equations for j = 2,..., JJ 1 Tis completes te construction of te block tridiagonal system at j interior points. For eac fixed j = 2,..., JJ 1, and fixed, U and V, we solve te following block tridiagonal system E n+ 1 2 z n ,j + F 1,j z n ,j = G n ,j z n 1,j 1 + Hn+ 1 2 z n 1,j G n ,j z n 1,j β n ,j, A n ,j z n ,j + B n+ 1 2 z n ,j + A n ,j z n ,j = C n ,j z n 2,j 1 + Dn+ 1 2 z n 2,j C n ,j z n 2,j+1 + βn ,j, A n ,j z n ,j + B n+ 1 2 z n ,j + A n ,j z n ,j = C n ,j z n 3,j 1 + Dn+ 1 2 z n 3,j (10.10) C n ,j z n 3,j+1 + βn ,j,.. F II,j z n+ 1 2 II 1,j + En+ 1 2 z n+ 1 2 II,j = G n+ 1 2 II,j zn II,j 1 + Hn+ 1 2 z n II,j G n+ 1 2 II,j zn II,j β n+ 1 2 II,j. Special boundary systems are constructed for te lines j = 1 and j = JJ. Tese systems are exactly constructed using boundary conditions and te details are given in Alemi Ardakani & Bridges (2009f ). Tis completes te algoritm details for te first alf step n n For eac fixed and U, it involves solving a sequence of linear block tridiagonal system for eac j = 1,..., JJ. Ten te process is repeated wit updates of and U till convergence n+ 1 2 and U U n Te second alf step is constructed similarly wit x derivatives explicit and y derivatives implicit, and te integration is along vertical grid lines. Te details are given in te report of Alemi Ardakani & Bridges (2009f ). 11. Numerical results: coupled roll-pitc forcing Suppose te vessel is prescribed to undergo a roll-pitc motion using te Euler angle representation in 8.1. Te roll and pitc motions are taken to be armonic and of te form φ (t) = ε r sin (ω r t) and θ (t) = ε p sin (ω p t).

23 Sallow water slosing in 3D 23 Figure 5. Snapsots of free surface profile at a sequence of times for coupled roll-pitc forcing. Te body representation of te angular velocity is ten ε r ω r cos (ω r t) cos(ε p sin ω p t) Ω(t) = ω p ε p cos ω p t ε r ω r cos (ω r t) sin(ε p sin ω p t), and te gravity vector is cos φ(t) sin θ(t) g(t) := gq T e 3 = g sin φ(t). cos φ(t) cos θ(t) Te vessel and fluid geometry parameters are set at L 1 = 1.0 m, L 2 = 0.80 m, 0 = 0.13 m, d 1 = 0.50 m, d 2 = 0.40 m, d 3 = 0.0 m. Te numerical parameters are set at x = 0.02 m, y = m, t = 0.01 s. Typical CPU time is about 5 seconds per time step witout any special optimisation, and a typical run is time steps. Simulations are run wit quiescent initial conditions: U = V = 0 and = 0 at t = 0. Wit tis fluid geometry, te natural frequencies (8.8) are ) 1/2 ω mn 3.54 (m 2 + n2 rad/sec,.64 wit te first two: ω rad/sec and ω rad/sec. Figures 5 and 6 sow snapsots of te free surface at a sequence of times wen ε p = 1.0, ε r = 2.0, and ω p = ω r = 2.40 rad/sec. Te forcing frequency is muc lower tan any of te natural frequencies in tis case, and te fluid

24 24 H. Alemi Ardakani and T. J. Bridges Figure 6. Snapsots of free surface in te roll-pitc case: continued. response is relatively gentle. Canging te roll and pitc excitation frequency to te natural frequency for eac direction, ω r = 3.54 rad/sec and ω p = 4.43 rad/sec, a steep bore forms at t = 2.0 sec as sown in Figure 7. Consider now te same amplitudes but a iger frequency ω r = ω p = 5.60 rad/sec, wic is close to te (1, 1) natural frequency. A bore is generated and its propagation between te tank s walls is illustrated in Figure 8. Te corresponding velocity fields associated wit Figure 8 are sown in Figure 9. As can be seen te water particles near te wave fronts ave greater speed tan in oter parts of te tank, and te numerical sceme captures te travelling ydraulic jump very well. 12. Yaw motion and potential vorticity conservation Te case of pure yaw forcing is of interest for several reasons. First, tere is a form of potential vorticity conservation; secondly, te surface SWEs and te HH SWEs agree in tis case (assuming equal velocity fields) and so te results can be compared, and tirdly tis case is te closest to te rotating SWEs in geopysical fluid dynamics. In te case of pure yaw forcing, te coefficients in te surface SWEs reduce considerably. Wit Ω 1 = Ω 2 = 0 and Ω 3 = ψ, were ψ(t) is te yaw angle, a 11 = a 22 = g, a 12 = a 21 = 0 b 1 = 2Ω 3 V + Ω 2 3(x + d 1 ) + Ω 3 (y + d 2 ) b 2 = 2Ω 3 U + Ω 2 3(y + d 2 ) Ω 3 (x + d 1 ). Te surface equations and te HH equations are identical (assuming equivalent velocity fields) and

25 Sallow water slosing in 3D 25 Figure 7. Snapsots of free surface profile in te roll-pitc case wen te forcing frequency is near te (1, 0) and (0, 10 natural frequencies. Figure 8. Roll-pitc forcing wit ig frequency, sowing formation and propagation of a ydraulic jump. te momentum equations reduce to te classical SWEs wit forcing U t + UU x + V U y + g x = 2 ψv + ψ 2 (x + d 1 ) + ψ(y + d 2 ) V t + UV x + V V y + g y = 2 ψu + ψ 2 (y + d 2 ) ψ(x + d 1 ). (12.1)

26 26 H. Alemi Ardakani and T. J. Bridges Figure 9. Field of velocity vectors associated wit te roll-pitc motion in Figure 8. Tese equations conserve a form of potential vorticity. Let Ten differentiating and using (12.1) sows tat D orz PV := V x U y + 2 ψ PV := t PV + U x PV + V y PV = 0, and so PV is a Lagrangian invariant for te SWEs (12.1). Consider te yaw motion to be armonic wit parameters Set te vessel and fluid parameters at. ψ (t) = ε y sin (ω y t), (12.2) ε y = 10.0, and ω y = rad/sec. L 1 = 1.0 m, L 2 = 1.0 m, 0 = 0.18 m, d 1 = 0.5 m, d 2 = 0.5 m, d 3 = 0.3 m. Te numerical parameters are set at x = y = 0.02 m, and t = 0.01 s. Te forcing frequency is near te lowest natural frequency. Te natural frequencies are ω mn m 2 + n 2 rad/sec. Snapsots of te surface profile at different values of time are depicted in Figures 10 and 11. Tere is a very clear swirling motion set up. Te corresponding velocity fields associated wit Figure 11 are depicted in Figure 12. Here we start to see evidence of a vorticity field. It is predominantly due to te input into te vorticity by te yaw rotation, but tis vorticity is enanced due to te presence of viscous-like truncation error in te numerical sceme.

27 Sallow water slosing in 3D 27 Figure 10. Snapsots of surface profile due to yaw forcing. Figure 11. Snapsots of surface profile due to yaw forcing: continued Comparison wit results of Huang & Hsiung (1996) Now cange te parameters in order to compare wit te results of Huang & Hsiung (1996). Set te vessel and fluid parameters at L 1 = 1.0 m, L 2 = 0.8 m, 0 = 0.1 m, d 1 = 0.5 m, d 2 = 0.4 m, d 3 = 0.0 m.

28 28 H. Alemi Ardakani and T. J. Bridges Figure 12. Velocity fields associated wit Figure 11. Te numerical parameters are x = 0.02 m, y = m, t = 0.01 s. Te forcing is armonic yaw motion as in (12.2) wit parameters ε y = 4.0, ω y = 6.0 rad/sec. Te snapsot of te surface profile at t = 1.5 s and te corresponding velocity field are depicted in Figure 13. Tis result agrees very well wit Figure 28 of Huang & Hsiung (1996), including te vortex pattern. Te initial conditions are vorticity free. Hence te generation of vorticity is due to te imposed rotation, but it appears to be enanced by te numerical dissipation. 13. Numerical results: coupled surge-sway motion Suppose te rotation is zero and te motion undergoes pure translation in te x and y directions. Tese surge and sway motions are considered to be armonic wit q 1 (t) = ε 1 cos (ω 1 t) and q 2 (t) = ε 2 cos (ω 2 t), ε 1 = ε 2 = m, ω 1 = ω 2 = rad/sec. Set te vessel and fluid aspect ratio parameters at L 1 = 0.59 m, L 2 = 0.59 m, 0 = m. Te offset d = 0 since tere is no rotation. Te numerical parameters are set at x = y = m, t = 0.01 s. Te forcing frequency is cosen to be very near te natural frequency of te exact (not SWE) lowest natural frequency ω gπ tan(π 0 ).

29 Sallow water slosing in 3D 29 Figure 13. Surface profile and velocity field due to yaw at t = 1.5 s: comparison of te numerics based on te surface SWEs wit Figure 28 of Huang & Hsiung (1996). Figure 14. Location of te points P 1 to P 8 in te tank cross section. Particular points are cosen on te vessel cross-section in order to sow time istories. Tis approac is inspired by te simulations of Wu & Cen (2009). Te points are labelled as sown in Figure 14. Te slosing istories of surface displacements at points P 1, P 2, P 4 and P 7 are depicted in Figure 15. Te parametric graps are depicted in Figure 16 sowing te presence of square-like waves in te tank.

30 30 H. Alemi Ardakani and T. J. Bridges Figure 15. Slosing istories of surface displacements at points P 1 (uppermost plot), P 2, P 4 and P 7 (lowest plot). Te orizontal axis is time in seconds, and vertical axis is wave eigt in meters. Figure 16. Parametric graps. and Canging te forcing parameters to ε 1 = m, ε 2 = m, ω 1 = ω 2 = rad/sec. Tese frequencies are close to lowest natural frequency of te full system (rater tan te SWE natural frequency) ω gπ tan(π 0 ).

31 Sallow water slosing in 3D 31 Figure 17. Time istories of te surface displacements at points P 1 (upper plot), P 2, P 4 and P 7 (lower plot). Te orizontal axis is time in seconds, and vertical axis is wave eigt in meters. Figure 18. Parametric graps. Te time istories of surface displacements at points P 1, P 2, P 4 and P 7 are depicted in Figure 17. Furter parametric graps are depicted in Figure 18 sowing te presence of planar waves in te tank. Snapsots of te surface profile at different times are depicted in Figures 19 and Numerical results: coupled yaw-pitc-roll motion Te vessel motion is simulated using te Euler angles introduced in 8.2. Te expression for Q is given in equation (8.6) and te required body represenation of te angular velocity is given in

32 32 H. Alemi Ardakani and T. J. Bridges Figure 19. Snapsots of surface profile due to surge and sway. Figure 20. Snapsots of surface profile due to surge and sway: continued. (8.7). Te yaw, pitc and roll angles are taken to be armonic ψ (t) = ε y sin (ω y t), θ (t) = ε p sin (ω p t), φ (t) = ε r sin (ω r t).

33 Sallow water slosing in 3D 33 Figure 21. Snapsots of surface profile due to roll, pitc and yaw. Te gravity vector is Set te fluid and vessel geometry parameters at sin θ(t) g(t) := gq T e 3 = g sin φ(t) cos θ(t). cos φ(t) cos θ(t) L 1 = 0.50 m, L 2 = 0.50 m, 0 = 0.12 m, d 1 = 0.25 m, d 2 = 0.25 m, d 3 = 0.0 m. Wit tis geometry te natural frequencies are ω mn 6.82 m 2 + n 2 rad/sec. We will present results for a typical run in tis configuration. Take te forcing function parameters to be ε y = 2.0, ε p = 1.0, ε r = 1.0, ω y = ω p = ω r = rad/sec, and set te numerical parameters at x = y = 0.01 m and t = 0.01 s. Wit tis low amplitude of forcing te singularity of te Euler angles is safely avoided. About 5.4 seconds of CPU time per required per time step, and simulations were run for about 600 time steps. Snapsots of te surface profile at a sequence of times are depicted in Figures 21 to Concluding remarks A new set of sallow water equations wic model te tree-dimensional rigid body motion of a vessel containing fluid as been derived. Te only assumptions are on te vertical velocity and acceleration at te surface. Te equations give new insigt into sallow water slosing, include te effect of viscosity, and numerical simulations are muc faster tan te full 3D equations. Te equations capture many of te features of 3D slosing for te case wen te free surface is single valued. In tis paper te vessel motion as been prescribed. Te vessel motion can also be determined by