Dynamic coupling between shallow-water sloshing and horizontal vehicle motion

Transcription

1 Under consideration for publication in Euro. Jnl of Applied Mathematics 1 Dynamic coupling between shallow-water sloshing and horizontal vehicle motion HAMID ALEMI ARDAKANI and THOMAS J. BR IDGES Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK H.Alemiardakani@surrey.ac.uk & T.Bridges@surrey.ac.uk (Received 4 November 29) The coupled motion between shallow water sloshing in a moving vehicle and the vehicle dynamics is considered, with the vehicle dynamics restricted to horizontal motion. The paper is motivated by Cooker s experiments and theory for water waves in a suspended container. A new derivation of the coupled problem in the Eulerian fluid representation is given. However, it is found that transformation to a Lagrangian representation leads to a formulation which has nice properties for numerical simulation. In the Lagrangian representation, a simple and fast numerical algorithm with excellent energy conservation over long times, based on the Störmer-Verlet method, is implemented. Numerical simulations of the coupled dynamics in both the linear and nonlinear case are presented. 1 Introduction The effect of liquid sloshing on the dynamics and control of liquid transport, e.g. the terrestrial transport of liquids, oil and liquid natural gas in ships, and fuel in aviation and astrodynamics, has motivated a wide range of research. An indication of the breadth of research in this area is the book of Ibrahim [11] which cites over 2 references. The interest in this paper is in the dynamic coupling between shallow-water fluid sloshing and the motion of the transport vehicle. Examples of where dynamic coupling is of interest is the sloshing of water on the the deck of fishing vessels [4], transport of liquid by robots [2, 21], motion planning for industrial control [9, 19, 8, 7], sloshing in automobile fuel tanks [23], and motion of water waves in a suspended container [6]. It is the latter paper that motivates the present study. In this paper, attention will be restricted to the case of horizontal motion only (for translation and rotation of the vessel coupled to shallow-water sloshing see [2]). Equations for this coupled problem have been derived by Cooker [6]. Let q(t) be the horizontal position of the vessel, then the governing equations are on the interval x L with h t + uh x + hu x = u t + uu x + gh x = q, u(, t) = u(l, t) =, for all t. The free surface is represented by the time-dependent curve y = h(x, t) and u(x, t) is a representative horizontal velocity. The equation for q(t) is Cooker s equation (1.1) q + νq = 1 2 ρg ( h(l, t) 2 h(, t) 2). (1.2) The coupling is nonlocal since the right-hand side of (1.2) can be interpreted as an integral 1 2 ρg ( h(l, t) 2 h(, t) 2) = ρg hh x dx. With an appropriate change of notation (1.2) is equation (9) in [6]. Here and throughout the paper ρ is the fluid density, which is taken to be constant. The fluid and vessel are assumed to have unit width. The mass of the dry vessel is denoted by and ν > is a spring constant. In [6] ν is related

2 2 H. Alemi Ardakani & T. J. Bridges Y y ν q (t) x Figure 1. Schematic of a moving vehicle constrained by a spring force partially filled with fluid. X to the tension in the suspension cables of the vessel. A schematic of the system is shown in Figure 1. A new derivation of (1.1)-(1.2) identifying the key assumptions is given in 2. When q(t) is given, equation (1.2) is neglected and the shallow water equations (1.1) are forced by the time-dependent function q(t). This problem has been extensively studied (e.g. [22, 5, 17, 18, 13]), and it has been shown to lead to very complicated motion. One measure of the difference with the coupled case is the role of energy. In the case of forced motion, the energy is not conserved where E(t) = de dt = q ρhu dx, ( 1 2 ρhu ρgh2 ) dx. On the other hand the energy of the coupled problem is conserved: dec dt E c (t) = = where ( 1 2 ρh(u + q) ρgh2 ) dx q νq2. (1.3) Previous work on the problem of coupled vehicle translation and sloshing includes [15, 12] where the coupled problem allowing for two dimensional flow is studied. In [15] the linear coupled problem is considered, and in [12] asymptotic results are derived. In [6] the fluid is considered to be shallow, and a detailed study of the adjusted natural frequencies is presented and compared with experiments. The aim of this paper is to develop a numerical method for the nonlinear coupled system (1.1)- (1.2) and present simulations. Solving the coupled system numerically would appear on the face of it to be straightforward. The system (1.1) can be solved by a typical shallow-water equation solver of which there are many, and the vessel equation (1.2) is just a forced harmonic oscillator. However, the fact that the acceleration (rather than velocity or position) of the vessel appears in (1.1) and nonlinear nonlocal coupling appears in (1.2) does present problems. At first we solved (1.1) using an implicit shallow-water equation solver. The scheme we used is the same one that is used in [1]. It is a very effective numerical scheme for the forced problem [1]. We coupled it with a standard fourth-order Runge-Kutta method for the vessel equation (1.2) and iteration was used to deal with the nonlinearities. This coupled numerical scheme is effective for waves which come close to breaking because the implicit scheme has a dissipative interface. However, in order to model long time oscillatory behaviour a scheme which conserves energy is of interest. Re-thinking the problem led us to consider the Lagrangian particle path (LPP) formulation for the fluid. In computational fluid dynamics, the Eulerian fluid representation is preferred over the Lagrangian representation since mesh distortion can cause problems. However in one space dimension mesh distortion is not a problem. Indeed, the LPP representation of the equations takes an elegant and simple form. Let x(a, t) be the position of a fluid particle, with a L the reference space. Then u = x t and the two equations in (1.1) reduce to one 2 x t 2 + g x a a ( ) χ = q. (1.4) x a

3 Coupled liquid-vehicle dynamics 3 Assuming nondegeneracy of the label map, x a, the wave profile is obtained from h = χ(a)x 1 a, with χ(a) := hx t= a determined by the initial conditions. The equation for q(t) in (1.2) remains the same with h expressed in terms of x(a, t). A remarkable feature of the coupled LPP equations is that they are completely determined by the Euler-Lagrange equation of the Lagrangian 1 functional L (x, q) = t2 t 1 ( 1 2 (ẋ + q)2 1 2 g χ(a) x a ) ρχ(a) dadt + t2 t 1 ( 1 2 q νq2) dt. (1.5) A derivation is given in 4.1. The advantage of the variational principle is that it is much easier to design a numerical scheme with good energy conservation properties. The Euler-Lagrange equations also have a Hamiltonian formulation. This structure suggests that a symplectic integrator can be used. In this paper a variant of the Störmer-Verlet algorithm is used. This algorithm has excellent energy and momentum conservation properties [14, 1], and it is explicit. Simulations are shown for both the linear and nonlinear case. In the linear case, away from resonance (see 5 for definition of resonance) an exact solution can be computed and comparison with the simulation shows excellent agreement. At resonance, simulations show distinctly non-harmonic motion. The nonlinear simulations show quite regular behaviour over a range of parameter values, although the initial conditions are restricted to a quiescent fluid. The initial momentum or position of the vehicle is prescribed, the latter being the initial condition in Cooker s experiments [6]. A numerial simulation with ν = is also presented showing vehicle drift coupled with vehicle wobble. 2 Governing equations There are two frames of reference. The spatial (inertial) frame has coordinates X = (X, Y ) and the body frame has coordinates x = (x, y). The whole system has a uniform translation, denoted q(t), in the x direction; hence X = x + q and Y = y. The vessel is a rigid body and the body frame is attached as shown in Figure 1. The fluid occupies the region y h(x, t) with x L. Assuming inviscid flow with velocity field (u(x, y, t), v(x, y, t)) and pressure field p(x, y, t), the Eulerian representation of the momentum equations for the fluid in the vessel relative to the body coordinate system is Du Dt + 1 p ρ x = q and Dv Dt + 1 p = g, (2.1) ρ y where g > is the gravitational constant and Du u Dt := t + u u x + v u y. Conservation of mass in the Eulerian representation, relative to the body frame, takes the usual form The boundary conditions at the vessel walls are u x + v y =. (2.2) u = at x = and x = L, v = at y =. (2.3) Neglecting surface tension, the boundary conditions at the free surface are The surface velocity field is defined by p = and h t + uh x = v, at y = h(x, t). (2.4) U(x, t) := u(x, h(x, t), t) and V (x, t) := v(x, h(x, t), t). 1 Note that the word Lagrangian is used in two distinct ways in this paragraph.

4 4 H. Alemi Ardakani & T. J. Bridges The horizontal surface velocity field satisfies the exact equation ( U t + UU x + g + Dv h) h x = q, (2.5) Dt To verify, differentiate the horizontal surface velocity h h h h U t + UU x = u t + u y h t + u(u x + u y h x ) = Du h h Dt + u y (h t + Uh x V ) = Du Dt Hence evaluation of the horizontal component of the momentum equation (2.1) at the free surface gives U t + UU x + 1 h ρ p x = q. But, remarkably, p x h ( = ρ g + Dv h) h x, (2.6) Dt h where Dv Dt is the Lagrangian vertical acceleration at the free surface. This latter identity follows from integrating the vertical momentum equation in (2.1), h Dv p(x, y, t) = ρg(h y) + y Dt ds, differentiating with respect to x and taking the limit y h. A detailed derivation of the identity (2.6) is given in [1, 3]. To derive an equation for h(x, t) start with the kinematic condition and recast it into the form h t + Uh x = V, h t + (hu) x = V + hu x. (2.7) This equation is exact, and neglect of the right-hand side reduces it to the form needed in the shallow water approximation. Equations (2.5) and (2.7) are the starting point for a one-dimensional analysis of the fluid motion. h. 2.1 Vehicle motion If the only force acting on the vehicle is the spring force, then Newton s second law gives ( d L ) h ρu dydx + (m f + ) q = νq, (2.8) dt where m f is the mass of the fluid: m f = h ρ dydx = ρh L with h = 1 L h(x, t) dx. The still water level h is independent of time because h(x, t) dx is a constant of the motion (for both the two-dimensional problem and the shallow water approximation). A more familiar form of the motion equation, involving the pressure, can be derived using the governing equations. Differentiating

5 Coupled liquid-vehicle dynamics 5 the total fluid momentum and using the horizontal momentum equation in (2.1), d dt h ρu dydx = = h h = m f q = m f q ρ Du Dt dydx ( p x ρ q) dydx ( L h x h p dy x=l x=, p dy ph x h ) dx where Reynold s transport theorem is used in the first line and the boundary condition p h = is used in the last line. Combining this equation with (2.8) gives q + νq = h p(l, y, t) dy h p(, y, t) dy. This latter equation is the one most often used in the literature (e.g. equations (3)-(4) in [12], and equation (27) in [15]). However, we will find that it is the form (2.8) that will be useful in the numerical simulation. To construct a shallow-water approximation for the coupled motion, first note that h ρudydx = ρhu dx h Substitute into the governing equation (2.8) ( ) d L ρhu dx + (m f + ) q + νq = dt h ρyu y dydx. ρyu y dydx, (2.9) 3 Coupled shallow-water sloshing and vehicle motion The exact equations for (h, U, q) are U t + UU x + gh x + q = Dv h Dt h x h t + (hu) x = V + hu x, ( ) d L h ρhu dx + (m f + ) q + νq = ρyu y dydx. dt (3.1) This set of equations is not closed since the right-hand sides contain terms involving v(x, y, t) and u(x, y, t). A closed set of equations for h(x, t), U(x, t) and q(t) is obtained by neglecting the right-hand sides of (3.1) h Dv L h Dt h x, V + hu x and ρyu y dydx. With these assumptions, the shallow water equations for the fluid are h t + hu x + Uh x = U t + UU x + gh x = q, (3.2) coupled to ( ) d L ρhu dx + (m f + ) q + νq =. (3.3) dt

6 6 H. Alemi Ardakani & T. J. Bridges This completes the derivation of (1.1)-(1.2). To see that (3.3) is the same as Cooker s equation (1.2), use (3.2) to transform d dt ρhu dx = = = m f q ρ(h t U + hu t ) dx ρ [ (hu 2 ) x + ghh x + h q) ] dx ρghh x dx. Substitution into (3.3) then gives (1.2). Cooker [6] shows that the linear problem has two important dimensionless parameters R = m f and G = νl2 4gh m f, (3.4) and they are equally important for the nonlinear problem. The other key dimensionless parameter is the fluid aspect ratio h /L. The use of the velocity at the free surface U(x, t) is one possible choice for the one-dimensional horizontal velocity. Another interesting choice is the vertical average of the two-dimensional horizontal velocity. This choice leads to the same equations (1.1)-(1.2) but with different assumptions. The implications of this latter choice are explored in Appendix A. 4 Lagrangian particle path formulation of the SWEs To derive the LPP form of the equations consider the mapping (t, a) (t, x(a, t)), with a L, t. Assuming non-degeneracy (x a ) the derivatives in (1.1) are mapped to Substitute into (1.1) t = t x tx 1 a a h t x t x 1 a and x = x 1 a a. h a + Ux 1 a h a + hx 1 a U a = U t x t x 1 a U a + Ux 1 a U a + gx 1 a h a = q. Setting U = x t and multiplying the first equation by x a simplifies these equations to Integrating the first equation x a h t + hx at = d dt (hx a) = x tt + gx 1 a h a = q. hx a = χ(a), where χ(a) := hx t= a is determined by the initial data. The second equation of (4.1) then reduces to 2 x t 2 + g ( ) χ = q. x a a This completes the derivation of (1.4). The equation for the vessel motion also needs to be transformed to the LPP setting. The total momentum transforms as ρhu dx = x a ρhux a da = ρuχ(a) da. Therefore Newton s law for the vessel motion in the LPP setting is ( ) d 2 L dt 2 ρx(a, t)χ(a) da + ( + m f )q(t) + νq(t) =, (4.2) (4.1)

7 and the LPP version of Cooker s equation is Coupled liquid-vehicle dynamics 7 q + νq = 1 2 m g f L χ(a) 2 x 2 a The first term in the brackets in (4.2) is proportional to the horizontal centre of mass (see Appendix C for derivation) and so the equation can be expressed in the form a=l a=. d 2 dt 2 (c.m. + (1 + R)q(t)) = ν m f q(t). (4.3) When the spring force is absent the position of the vehicle is principally determined by the position of the centre of mass of the fluid. 4.1 Lagrangian variational principle for the coupled problem The coupled equations for x(a, t) and q(t) can be determined from a variational principle using the Lagrangian functional introduced in (1.5). Taking the first variation of the Lagrangian action (1.5) with respect to q gives d ds L (x, q + s q) = s= = t2 t 1 t2 t 1 t2 q t (x t + q t ) ρχ(a)dadt + ( q t q t ν qq) dt, [ t 1 ] L t2 q t x t ρχ(a)da + (m f + )q t dt ν qqdt. t 1 t 1 Integrating by parts and using fixed endpoint conditions on the variations q gives the equation for q in (4.2). Now take the first variation of the action integral (1.5) with respect to x, t2 ( d ds L (x + s x, q) = (x t + q t ) x t + gχ(a) ) s= t 1 2x 2 x a ρχ(a)dadt a t2 ( = x tt q tt gχ a x 2 + gχx ) aa a x 3 xρχ(a)dadt, a where integration by parts has been used to obtain the second equation, with boundary terms neglected. Setting this expression to zero recovers the LPP equation for x(a, t) in (1.4). We have also found a variational principle for the Eulerian form of the coupled equations (see Appendix B), but it is (so far) less useful for constructing a numerical scheme. 4.2 Hamiltonian formulation The coupled system can also be expressed in Hamiltonian form with canonical variables (q, x, p, w). It is a mixed system in the sense that x(a, t), w(a, t) depend on a as well as t but q(t), p(t) are dependent on t only. The Hamiltonian formulation will be useful when we choose a numerical method for the simulations. It is derived by taking the Legendre transform of L. The resulting momentuariables are w(a, t) = x t (a, t) + q t (t) p(t) = ( + m f )q t (t) + x t(a, t)ρχ(a)da. The Hamiltonian functional is H(x, q, w, p) = 1 2 w2 ρχ da 1 p ( ) L 2 2 wρχ da + 1 wρχ da νq2 + 1 p 2 1 χ2 2ρg x a da. (4.4) (4.5)

8 8 H. Alemi Ardakani & T. J. Bridges The governing equations in Hamiltonian form are ẇ = δh δx = gχa x gχ 2 a x x 3 aa a ṗ = δh δq = νq ẋ = δh δw = w 1 p + 1 w ρχ(a)da q = δh δp = 1 p 1 w ρχ(a)da. (4.6) 5 Linear coupled problem An analysis of the linear problem will be useful for determining how the natural frequencies of the fluid sloshing are modified by the coupling. The linear natural frequencies obtained by [6] in the Eulerian setting will be recovered here in the LPP setting. Let x(a, t) = a + X(a, t) and h(a, t) = h + H(a, t), and linearize the governing equations about the trivial solution. Then χ(a) = h and X(a, t) and H(a, t) satisfy with boundary conditions X(, t) = X(L, t) = and and the linearized vehicle motion equation is ( d 2 ) m L f dt 2 X(a, t) da + ( + m f )q(t) L Consider solutions that are periodic in time of frequency ω, X tt gh X aa = q tt, (5.1) H + h X a =, (5.2) X(a, t) = X(a) sin ωt h(a, t) = Ĥ(a) sin ωt q(t) = q sin ωt. Substitution into (5.1)-(5.3) results in the coupled integro-differential system X aa + α 2 X = α2 q, + νq =. (5.3) (5.4) Ĥ + h Xa =, (5.5) where m f L X(a) da + (m f + ) q = ν ω 2 q, α = ω gh, (5.6) and the boundary conditions are X() = X(L) =. The first two equations of (5.5) can be solved for q, X and Ĥ, q = C 1 cot 1 αh 2αL, (5.7) where C represents an arbitrary nonzero multiplicative constant, and A key assumption in this derivation is X(a) = C 1 αh ( cot 1 2 αl(1 cos αa) sin αa), (5.8) Ĥ(a) = C ( cos αa cot 1 2 αl sin αa). (5.9) sin 1 2αL. (5.1)

9 Coupled liquid-vehicle dynamics 9 The implications of this singularity are discussed below. Substituting the expressions (5.7)-(5.9) into the third equation in (5.5) and using the integral gives where or since C, X(a) da = C 2 α 2 h ( 1 2 αl cot 1 2 αl 1), 2ρ α 2 C(s cot s 1) m C f cot s = ( νω 2 ) C cot s, αh αh s = 1 2 αl, 2 g L m f αh = ( ω 2 + ν) cot s. To see that this agrees with Cooker s result recast in terms of the dimensionless parameters R and G in (3.4). Then (s) =, with (s) = G Rs tan s, (5.11) s which is precisely equation (15) in [6]. Cooker shows that equation (5.11) has a countable number of solutions, s j with j = 1, 2,.... Moreover, for fixed R, G and s > the derivative satisfies (s) <. Hence all roots of (s) = are simple. Given a root s j satisfying (s j ) =, the natural frequency of the coupled system is then given by ω coupled j = 2 L gh s j. (5.12) 5.1 Natural frequencies For comparison, the natural frequencies of the uncoupled problem are recorded. The fluid natural frequency when the tank is fixed (the sloshing frequency) is ω f j := 2 gh L The natural frequency of the dry vessel is jπ 2, j = 1, 2,.... ω v = ν gh G = 2 L R. The natural frequency of the dry vessel plus fluid with the fluid treated as rigid body is ν gh G ω fv = = 2 m f + L (1 + R). Cooker shows that ω coupled 1 is in fact strictly less than all other natural frequencies. ω coupled 1 < ω f 1, ωcoupled 1 < ω v, ω coupled 1 < ω fv. (5.13) The first inequality follows from the fact that s 1 < 1 2π [6]. Furthermore, ωcoupled 1 ω f 1 only in the limit G. The third inequality in (5.13) is verified as follows. Use the inequality tan(s 1 ) > s 1 (since 2s 1 < π) and then G G Rs 1 = tan s 1 > s 1 s 1 < s R ωcoupled 1 < ω fv. A similar argument confirms the second inequality in (5.13). The singularity (5.1) can be expressed in terms of natural frequencies sin 1 2 αl = 1 2 αl = jπ ω = 2 gh L jπ.

10 1 H. Alemi Ardakani & T. J. Bridges But when s j = jπ then tan s j = and so (5.11) reduces to G jπ Rjπ = or G R = j2 π 2, j = 1, 2,.... Cooker refers to this as resonance, since G = j 2 π 2 R implies that The main consequence of the singularity ω coupled j = ω v and ω v = 2ω f j. is that the class of periodic solutions (5.4) no longer exists. G R = j2 π 2, j = 1, 2,..., (5.14) 6 Numerical algorithm using the Störmer-Verlet method Discretize the reference space by letting a i = (i 1) a, i = 1,..., N + 1, with a = L N, and let x i (t) := x(a i, t) and w i (t) := w(a i, t). Then after choosing a discretization for the derivatives with respect to a, the governing equations can be written as a large set of ordinary differential equations of the form ṗ = g(q) q = f(p), for some functions f( ) and g( ) whose precise form will be made clear shortly, and where p = (p, w 1,..., w N+1 ) and q = (q, x 1,..., x N+1 ). The important point here is the fact that the right hand side of the ṗ equation depends only on q and the right hand side of the q equation depends only on p. There are appealing numerical methods for equations of this form: split-step methods, or partitioned Runge-Kutta methods. The simplest of this class of methods is the discretization p n+ 1 2 = p n t g (qn ) ( ) q n+1 = q n + t f p n+ 1 2 p n+1 = p n t g ( q n+1). This scheme is explicit, and has second-order accuracy in time. If in addition the equations are Hamiltonian as in (4.6) then this scheme is called the Störmer-Verlet method and it has additional properties [14, 1]. It is symplectic, and has excellent energy conservation over long time intervals. To apply this method to the equations (4.6) is straightforward. There are only two tricky points: how to discretize the right-hand side of the first equation in (4.6), and secondly how to discretize the integral term in the third and fourth equations. For the first difficulty, there are two obvious ways of discretizing: directly discretize both terms or discretize it as ( ) g χ. x a x a To decide which, there is a more natural way to approach this problem. It is now known that when an equation is an Euler-Lagrange equation often the best way to discretize is to discretize the Lagrangian functional first, and then take a variation of the discretized Lagrangian [16]. In this case the functional that generates the term is F (x) = a gχ(a) 2x a ρχ(a)da.

11 Discretize using a finite difference formula for the derivative Coupled liquid-vehicle dynamics 11 F (x 1,..., x N+1 ) = Now take a variation in (x 1,..., x N+1 ), N 1 gχ i a 2 (x i+1 x i ) ρχ i a. i=1 d N g ds F (x 1 + sξ 1,..., x N+1 + sξ N+1 ) = s= 2χ i A semi-discretization of the first equation in (4.6) is then ẇ = g a ( χ 2 i 1 2χ i (x i x i 1 ) 2 χ 2 ) i (x i+1 x i ) 2 For the integral of w that appears in (4.6) Simpson s rule is used σ n+ 1 2 := wn+ 1 2 ρχda 1 3 (w ρ a n χ N i= i=1 wn+ 2 ( χ 2 i 1 a (x i x i 1 ) 2 χ 2 i a ) (x i+1 x i ) 2 ρξ i χ i a. 2i+1 χ 2i N 2, i = 2,..., N. i=1 wn i χ 2i + w n+ 1 2 N+1 χ N+1 The above discretization combined with the Störmer-Verlet method then gives the following equations for each time step n n + 1 ). where p n+ 1 2 = p n t 2 νqn, w n+ 1 2 i = w n i + g t a ( χ 2 i 1 4χ i (x n i xn i 1) 2 χ 2 i (x n i+1 xn i ) 2 x n+1 i = x n i + 1 twn+ 2 i t p n t σ n+ 1 2, q n+1 = q n + t p n+ 1 2 t σ n+ 1 2, p n+1 = p n+ 1 2 t 2 νqn+1, w n+1 i = w n+ 1 2 i + g t a ( χ 2 i 1 4χ i (x n+1 i x n+1 i 1 ) 2 ), i = 2,..., N, i = 2,..., N χ 2 i (x n+1 i+1 xn+1 i ) 2 ), i = 2,..., N, w n i = w (a i, t n ), x n i = x (a i, t n ), p n = p (t n ), q n = q (t n ), t n = n t, and the boundary conditions are x n+1 1 =, x n+1 N+1 = L, w n = w n+ 1 2 N+1 = 1 (+m f ) pn+ 1 2, w n+1 1 = w n+1 N+1 = 1 (+m f ) pn+1. (6.1) 6.1 Energy conservation The energy (1.3) of the coupled system, transformed to the LPP setting is ( 1 E (t) = 2 (U + q) g χ ) ρχ(a)da + 1 x a 2 q νq2. (6.2) This energy is conserved by the coupled equations (1.4)-(4.2). Indeed when transformed to canonical coordinates (q, x, p, w) it is the Hamiltonian function (4.5). Using the canonical variables, an approximation of the integrand in (6.2) at a = a i is E i = 1 2 ρχ iw 2 i ρgχ ih i.

12 12 H. Alemi Ardakani & T. J. Bridges Using Simpson s rule the discrete approximation of the energy in the LPP setting becomes E a 3 (E N 2 1 i=1 E 2i ) N 2 i=1 E 2i + E N q νq2. (6.3) One of the remarkable properties of the Störmer-Verlet algorithm is that the energy error is O( t 2 ) uniformly in time. This property is proved in [1]. In other words the magnitude of the error does not increase in time. This property is observed in the numerics and is shown in the results in 7 and 8. However, our numerics show that the energy error is O( t p ), with 1 < p < 2, uniformly in time. This larger error estimate is probably due to the fact that we are applying the Störmer-Verlet algorithm to a PDE, and there is additional error due to numerical integration using Simpson s rule in (6.3), whereas the proof in [1] is for ODEs only. However, the important property is that the energy error is uniform in time, hence long time simulations are possible. The linearized system is 7 Numerical simulations linear system ẇ = gh X aa, ṗ = νq, Ẋ = w 1 p + 1 L LR w da, q = 1 p 1 L LR w da. This system is also Hamiltonian with functional H(X, q, w, p) = 1 2 ρh w2 da ρh p w da p 2 ( + 1 ρ 2 h 2 ) L 2 2 w da νq2 + 1 L 2 ρgh2 X2 a da. The Hamiltonian functional is the energy of the linear system and in terms of the original variables it is ( ) E = 1 2 (X t + q t ) 2 + gh Xa 2 ρh da q νq2. (7.3) The system (7.1) is discretized using a centred finite-difference formula for X aa, Simpson s rule for the integral, and Störmer-Verlet for time integration. The boundary conditions are X(, t) = X(L, t) =, w(, t) = w(l, t) = 1 + m f p(t). The first simulation is a comparison with the exact oscillatory solution of 5. For this simulation, take initial data and for the momenta, X(a, ) =, and q() =, w(a, ) = ω q + ω X(a) and p() = ( + m f )ω q + ω m f L (7.1) (7.2) X(a) da. (7.4) Once the parameters are fixed, the frequency ω is obtained by solving (s) = using Newton s method, and then substituting the value of s into (5.12). The chosen values of the parameters are listed in Table 1. The values are guided by the experiments in [6]. The spring stiffness in [6] is determined by the tension in the suspension cables ν = g l ( + m f ), where l is the length of the suspension cable. Based on the range of values in the experiments, the range of ν is ( 1.3 < ν < kg/sec 2 ).

13 Coupled liquid-vehicle dynamics 13 Figure 2. Comparison of computed horizontal position and the exact solution given in (5.4), for two values of a. The right-hand figures show the distribution of Courant number, illustrating its affect on phase error. Moreover, the range of R and G in [6] are.75 < R < and.32 < G < Figure 2 shows the numerical and exact solutions and local Courant number at different values of time for the input data tabulated in Table 1. The position q(t) is shown for large times. The spring stiffness is ν = 5 kg/sec 2 and the frequency ω = ω coupled 2. The oscillatory behaviour and its frequency are simulated very well for long times. The only significant error is phase error, and the phase error is strongly affected by the local Courant number Cr (a, t) = t a X t (a, t). In the upper left plot in Figure 2 the phase error is prominent, but then making the step size smaller brings the average Courant number closer to unity and the phase error is largely eliminated. Increase the spring stiffness to ν = 17 kg/sec 2 and take ω = ω coupled 1. Figure 3 shows the numerical and exact solutions for this case, and in the lower plot the energy error is shown. The energy error is of order t, uniformly in time. Snapshots of the Eulerian wave profile for this case are shown in Figure 4. The complete set of input data is tabulated in Table 1. The Eulerian wave profile, h(x, t), is determined by plotting h(a, t) versus x(a, t) for fixed t, generating a curve parameterized by a. Now consider the case where ω = ω coupled 2, ν = 15 kg/sec 2 and G < π 2 R. With the input data as tabulated in the first column of Table 2, the numerical and exact solutions, energy error, and local Courant number at different values of time are shown in Figure 5. Figure 6 shows snapshots of the Eulerian wave profile for this case. The wave profile has the appearance of a nonlinear wave, but this is misleading. The solutions are linear in the Lagrangian setting. The wave height h(a, t) is harmonic as a function of a, but when it is mapped through x(a, t) it appears nonlinear. Now consider the case where ω = ω coupled 1, ν = 38 kg/sec 2 and G > π 2 R. With the input data tabulated in the second column of Table 2, the numerical and exact solutions, energy error, and local Courant number at different values of time are shown in Figure 7, and snapshots of Eulerian wave profile are shown in Figure 8. Consider the same parameter values as Figure 7 but take the frequency equal to the third natural frequency of the coupled problem. With the input data tabulated in the third column of Table 2, the numerical and ecaxt solutions, energy error, local Courant number at different values of time, and snapshots of Eulerian wave profile when ω = ω coupled 3 and G > π 2 R are shown in Figures 9 and 1.

14 14 H. Alemi Ardakani & T. J. Bridges Table 1. Input data for the numerical experiments in Figures 2 4. The fluid density is ρ = 1 kg/m 3, the gravitational constant is set at g = 9.81 m/s 2, the tank length is L =.525 m, and the still water level is h =.9 m. The time step in all cases is t = 1 4 and the number of time steps is typically 1 6. Figure 2 (first row) 2 (second row) 3 and 4 ν (kg/sec 2 ) m f (kg) (kg) C (m) ω (rad/sec) =ω coupled =ω coupled =ω coupled 1 h /L R G ω coupled 1 (rad/sec) ω f 1 (rad/sec) ω v (rad/sec) ω fv (rad/sec) a (m) t CP U (sec) Figure 3. Comparison of the computed horizontal position and the exact solution for long and short times, and the lower graph shows the energy error.

15 Coupled liquid-vehicle dynamics 15 Figure 4. Snapshots of Eulerian wave profile associated with Figure 3. The horizontal and vertical axes are, respectively, x(a, t) and h(a, t). Figure 5. Comparison of exact and computed vehicle horizontal motion, energy error, and local Courant number at different values of time when ω = ω coupled 2 and G < π 2 R. In Figure 1 a multi-peak wave shows up due to the fact that a higher mode is simulated. 7.1 Simulations near resonance Consider the case of of resonance, when G = j 2 π 2 R. The most interesting case is j = 1. In this case ω v is equal to ω coupled 1 and ω v = 2ω f 1. However the line G = j2 π 2 R is not a resonance in the usual sense since ω v and ω f 1 are not natural frequencies of the coupled problem. Nevertheless, this line in parameter space is of interest since the harmonic solution of 5 is no longer valid. In the resonance case the initial conditions (7.4) are no longer valid, and so the following initial

16 16 H. Alemi Ardakani & T. J. Bridges Figure 6. Snapshots of Eulerian wave profile associated with Figure 5. Table 2. Input data for the numerical experiments in Figures The fluid density ρ = 1 kg/m 3, the gravitational constant is set at g = 9.81 m/s 2, and the tank length is L =.525 m. In all cases the space step is a = m and time step is t = 1 4 sec, the number of time steps is typically 1 6 and the CPU time per simulation is typically 2 minutes. Figure 5 and 6 7 and 8 9 and 1 11 and and 14 q () =.4 q () =.15 h (m) ν (kg/sec 2 ) m f (kg) (kg) C (m) ω (rad/sec) = ω coupled 2 = ω coupled 1 = ω coupled 3 h /L R G < π 2 R > π 2 R > π 2 R = π 2 R = π 2 R ω coupled 1 (rad/s) ω f 1 (rad/sec) ω v (rad/sec) ω fv (rad/sec)

17 Coupled liquid-vehicle dynamics 17 Figure 7. Comparison of exact and computed vehicle horizontal motion, energy error, and local Courant number at different values of time when ω = ω coupled 1 and G > π 2 R. Figure 8. Snapshots of Eulerian wave profile associated with Figure 7. conditions are used h (a, ) = h, X (a, ) =, q() =, w (a, ) = q (), p () = ( + m f ) q (), with q () specified. Setting q () =.4, and with the other parameter values tabulated in Table 2, the results of numerical test for vehicle horizontal motion, energy error, and local Courant number at different values of time are shown in Figure 11 and snapshots of the Eulerian wave profile are shown in Figure 12. Although the solution looks periodic there are clearly higher harmonics in the solution. Instead of setting the initial velocity of the vehicle, assign an initial displacement of the vehicle, but

18 18 H. Alemi Ardakani & T. J. Bridges Figure 9. Comparison of exact and computed vehicle horizontal motion, energy error, and local Courant number at different values of time. Figure 1. Snapshots of Eulerian wave profile associated with Figure 9. with a quiescent fluid. The complete set of initial conditions is h (a, ) = h, X (a, ) =, q (), w (a, ) =, p () =, q () =. Set q () =.15, with the other parameter values as tabulated in Table 2. The results of the numerical test for vehicle horizontal motion, energy error, and local Courant number at different values of time are shown in Figure 13, and snapshots of the Eulerian wave profile are shown in Figure 14. This simulation shows that distinctly non-harmonic behaviour of the solution near resonance.

19 Coupled liquid-vehicle dynamics 19 Figure 11. Computed vehicle horizontal motion, energy error, and local Courant number at different values of time. Figure 12. Snapshots of Eulerian wave profile associated with Figure Numerical simulations nonlinear system Now consider simulation of the full nonlinear discretization (6.1). For the initial conditions the fluid is taken to be quiescent and nonzero initial conditions are applied to the vehicle. For the first simulation, take ν = 13 kg/sec 2, and set the initial velocity of the vehicle at q() =.1. The other parameter values are listed in the first column of Table 3. The complete set of initial conditions is h (a, ) = h, x (a, ) = a, q() =, q () =.1, w (a, ) = q (), p () = ( + m f ) q (). The computed vehicle horizontal motion, energy error, and snapshots of Eulerian wave profile are shown in figures 15 and 16. For the second nonlinear simulation, take ν = 35 kg/sec 2 and set the initial position of the vehicle

20 2 H. Alemi Ardakani & T. J. Bridges Figure 13. Computed vehicle horizontal motion, energy error, and local Courant number at different values of time. Figure 14. Snapshots of Eulerian wave profile associated with Figure 13. to be nonzero. Other parameter values are listed in the second column of Table 3. The complete set of initial conditions is h (a, ) = h, x (a, ) = a, q() =.5, q () =, w (a, ) =, p () =. The computed vehicle horizontal motion, energy error, and snapshots of Eulerian wave profile are shown in figures 17 and 18. The third numerical experiment is to give the free surface an initial displacement, with the fluid quiescent, and determine the motion of the vehicle when the spring force is neglected. The initial free surface position is ( ) L h (a, ) = h + 2 a tan θ. (8.1)

21 Coupled liquid-vehicle dynamics 21 Figure 15. Computed vehicle horizontal motion for long and short time, and energy error when q() = and q() =.1 associated with the first nonlinear simulation. Figure 16. Snapshots of Eulerian wave profile associated with Figure 15. In this case the function χ(a) is χ(a) := hx a t= = h (a, ) = h + ( ) L 2 a tan θ. With the input data tabulated in the last column of Table 3, the numerical solution for q (t) and the energy error are shown in Figure 19. There is a slow drift of the vehicle due to the lack of the

22 22 H. Alemi Ardakani & T. J. Bridges Table 3. Input data for the numerical experiments in Figures The fluid density ρ = 1 kg/m 3, the gravitational constant is set at g = 9.81 m/s 2, and the tank length is L =.525 m. The space step and time step in all cases are a = m and t = 1 4 s, the number of time steps is 1 6 and the CPU time per simulation is typically about 2 minutes. Figure 15 and and h (m) ν (kg/sec 2 ) m f (kg) (kg) q() (m/sec).1..1 q() (m)..5. h /L R G ω coupled 1 (rad/s) ω f 1 (rad/sec) ω v (rad/sec) ω fv (rad/sec) horizontal restoring force. When ν = the equation for q(t) can be integrated (see Appendix C), Substituting for the initial data gives x(a, t)ρχ(a)da + ( + m f ) q(t) = c 1 t + c. q(t) = q() t 1 L [x(a, t) a] χ(a) h da. (8.2) The first term gives a uniform drift. With q() =.1 m/sec the drift should amount to about 1 m after 1 seconds and this is observed in the upper graph in Figure 19. The second term in (8.2) gives a small wobble to the vehicle motion due to the oscillation of the fluid centre of mass. By zooming in on the graph of q(t) versus t for short times, this oscillation can be seen and is shown in the middle graph in Figure 19. The energy error is of order 1 6 for all time, which is O( t 3/2 ). When ν = the first coupled natural frequency, ω coupled 1, does not exist since G = and so (s) = Rs tan s which has no solutions in the interval < s < 1 2 π. 9 Concluding remarks Simulations of the coupled problem of shallow water sloshing and vehicle dynamics have been presented. Two of the central features of the model can be generalized. The linear spring can be generalized to a nonlinear spring. The linear spring force is νq and a nonlinear spring force would be of the form F = ν 1 q + ν 2 q 3,

23 Coupled liquid-vehicle dynamics 23 Figure 17. Computed vehicle horizontal motion for long and short time, and energy error when q() =.5 and q() = associated with the first nonlinear simulation. Figure 18. Snapshots of Eulerian wave profile associated with Figure 17. where ν 1 > and ν 2 are constants. If ν 2 < the spring is called hard and if ν 2 > it is called soft. With this spring force, Cooker s equation (1.2) is modified to q + ν 1 q ν 2 q 3 = 1 2 ρg ( h(l, t) 2 h(, t) 2). Hence the vehicle motion is determined by a forced Duffing oscillator. Since a forced Duffing oscillator can have very complicated dynamics, it is expected that the fluid sloshing will be more dramatic in this case.

24 24 H. Alemi Ardakani & T. J. Bridges Figure 19. Computed vehicle horizontal motion and energy error when ν =. and the initial surface profile is (8.1) with θ =.5 degrees. The vehicle motion was restricted to horizontal translation. However, vertical translation and rotation can also be considered, and this coupled problem is studied in [2]. Appendix Appendix A Exact equations based on the average horizontal velocity In deriving the shallow-water equations in 3 the velocity at the free surface is used for the onedimensional horizontal velocity. There are other possible choices. In this appendix it is shown that another interesting choice is the vertical average of the horizontal velocity. When the vertically averaged horizontal velocity u(x, t) = 1 h h u(x, y, t) dy, is used then two out of the three equations (3.2)-(3.3) for the coupled problem are exact. Differentiating ( ) (hu) x = h h h h h h u dy = h x u x + (u x + v y ) dy v y dy = h x u v = h t, using mass conservation, the kinematic free surface condition and the bottom boundary condition. Hence the equation for h is exact h t + (hu) x =.

25 Coupled liquid-vehicle dynamics 25 Similarly the equation for the vehicle motion is also exact since (2.8) is in terms of u, ( ) d L ρhudx + (m f + ) q = νq, dt However the momentum equation is not exact. Starting with the 2D horizontal and vertical momentum equations (2.1) one can verify by direct calculation that u t + uu x + gh x = q Rem, where the remainder has the form [ ( Rem = 1 h u 2 u 2 + h x h y ) ] Dv Dt ds dy. Hence to use u the only assumption required to close the equations is Rem 1. It is not clear when this term is small, but sufficient conditions are negligible vertical accelerations and u 2 u 2. Appendix B Variational principle for the coupled system when the fluid motion is Eulerian The governing equations in the Eulerian representation can also be obtained from a variational principle. Let L (q, h, U) = [ 1 2 ρh(u + q)2 1 2 ρgh2 + ρφ(h t + (hu) x ) ] dx q νq2. This Lagrangian functional (for the Eulerian flow field) is of the form where KE = L = KE P E + constraint, 1 2 ρh(u + q)2 dx q 2 and P E = 1 2 ρgh 2 dx νq2. with conservation of mass h t + (hu) x = imposed as a constraint. The Lagrange multiplier φ is a velocity potential, but since the equations are one-dimensional the potential does not imply potential flow. Take variations of this Lagrangian L φ = h t + (hu) x = L h = 1 2 (U + q)2 gh φ t φ x U = L u = h(u + q) φ xh = L q = d dt ( ρh(u + q)dx ) q νq =. The first equation is mass conservation. The third equation gives φ x = U + q. Substitute this expression into the second equation, φ t = 1 2 (U + q)2 gh (U + q)u φ t U 2 + gh = 1 2 q2, a form of Bernoulli s equation. Differentiate it which simplifies to φ xt = (U + q)u x gh x UU x (U + q)u x, U t + UU x + gh x = q,

26 26 H. Alemi Ardakani & T. J. Bridges recovering the momentum equation in (3.2). The variation L / q = recovers the equation for q(t) in (3.3). However this variational formulation does not lead to an effective numerical scheme as far as we are aware. Appendix C Case of spring constant zero If the spring constant ν is zero, then the LPP equation for the vehicle motion (4.2) reduces to ( ) d 2 L dt 2 ρxχ(a)da + (m f + ) q =, or xρχda + ( + m f ) q = c 1 t + c, (C-1) where c and c 1 are determined by the initial data. In this case the position of the vehicle can be expressed in terms of the fluid motion with q (t) = c 1 m f + t + c 1 L m f + m f + c 1 = ρu (a, ) da + (m f + ) q (), xρχda, (C-2) c = x (a, ) ρχda + (m f + ) q (). This result shows that the position of the vehicle q(t) is completely determined by the initial data, and the horizontal component of the centre of mass c.m. of the fluid, since c.m. = 1 m f h ρx dydx = 1 m f x(a, t)ρχ(a) da. References [1] Alemi Ardakani, H. & Bridges, T. J. 29 Shallow-water sloshing in rotating vessels. Part 1: Twodimensional flow field, Preprint: [2] Alemi Ardakani, H. & Bridges, T. J. 29 Coupled roll motion of a liquid-tank system with shallow water sloshing, (in preparation). [3] Bridges, T. J. 29 Wave breaking and the surface velocity field for three-dimensional water waves, Nonlinearity [4] Caglayan, I. & Storch, R. L Stability of fishing vessels with water on deck: a review, J. Ship Research [5] Chester, W Resonant oscillations of water waves. I. Theory, Proc. Royal Soc. London A [6] Cooker, M. J Water waves in a suspended container, Wave Motion 2, [7] Dubois, F., Petit, N. & Rouchon, P Motion planning and nonlinear simulations or a tank containing fluid, European Control Conference, Karlsruhe, 6 pages. [8] Feddema, J. T., Dohrmann, C. R., Parker, G. G., Robinett, R. D., Romero, V. J. & Schmitt, D. J Control for slosh-free motion of an open container, IEEE Control Systems Magazine [9] Grundelius, M. & Bernhardsson, B Control of liquid slosh in an industrial packaging machine, IEEE Int. Conf. Control Appl., Kohala Coast, Hawaii, 6 pages. [1] Hairer, E., Lubich, C. & Wanner, G. 23 Geometric numerical integration illustrated by the Störmer- Verlet method, Acta Numerica [11] Ibrahim, R. A. 25 Liquid Sloshing Dynamics, Cambridge University Press. [12] Ikeda, T. & Nakagawa, N Non-linear vibrations of a structure caused by water sloshing in a rectangular tank, J. Sound & Vibration [13] Jones, A. F. & Hulme, A The hydrodynamics of water on deck, J. Ship Research [14] Leimkuhler, B. & Reich, S. 24 Simulating Hamiltonian Dynamics, Cambridge University Press. [15] Lui, P.-C. & Lou, Y. K. 199 Dynamic coupling of a liquid-tank system under transient excitation, Ocean Engng [16] Marsden, J.E. & West, M. 21 Discrete mechanics and variational integrators, Acta Numerica

27 Coupled liquid-vehicle dynamics 27 [17] Ockendon, J. R. & Ockendon, H Resonant surface waves, J. Fluid Mech [18] Ockendon, H., Ockendon, J. R. & Johnson, A. D Resonant sloshing in shallow water, J. Fluid Mech [19] Prieur, C. & de Halleux, J. 24 Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations, Systems & Control Letters 52, [2] Tzamtzi, M. P. & Kouvakas, N. D. 27 Sloshing control in tilting phases of the pouring process, Inter. J. Math., Phys. Eng. Sciences [21] Tzamtzi, M. P. & Kouvakas, N. D. 28 Robustness of a robot control scheme for liquid transfer, in Novel Algortithms and Techniques in Telecommunications, Automation and Industrial Electronics, Edited by T. Sobh et al., Springer-Verlag [22] Verhagen, J. H. G. & van Wijngaarden, L Non-linear oscillations of fluid in a container, J. Fluid Mech [23] aus der Wiesche, S. 23 Computational slosh dynamics: theory and industrial application, Comp. Mech