Download Dynamic coupling between shallow-water sloshing and horizontal

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Under consideration for publication in Euro. Jnl of Applied Mathematics
Dynamic coupling between shallow-water sloshing
and horizontal vehicle motion
H A M I D A L EM I A R D A K A N I and T H O M A S J. B R I D GE S
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
email: [email protected] & [email protected]
(Received 4 November 2009)
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 2000 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 [20, 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
ht + uhx + hux
=
0
ut + uux + ghx
= −q̈ ,
(1.1)
on the interval 0 6 x 6 L with
u(0, t) = u(L, t) = 0 ,
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
mv q̈ + νq = 21 ρg h(L, t)2 − h(0, t)2 .
(1.2)
The coupling is nonlocal since the right-hand side of (1.2) can be interpreted as an integral
Z L
2
2
1
hhx dx .
ρg
h(L,
t)
−
h(0,
t)
=
ρg
2
0
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 mv and ν > 0 is a spring constant. In [6] ν is related
2
H. Alemi Ardakani & T. J. Bridges
Y
y
ν
x
q (t)
X
Figure 1. Schematic of a moving vehicle constrained by a spring force partially filled with fluid.
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 q̈(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
Z L
dE
ρhu dx ,
= −q̈
dt
0
where
Z
E(t) =
0
L
( 21 ρhu2 + 12 ρgh2 ) dx .
c
On the other hand the energy of the coupled problem is conserved: dE
dt = 0 where
Z L
Ec (t) =
( 21 ρh(u + q̇)2 + 12 ρgh2 ) dx + 12 mv q̇ 2 + 12 νq 2 .
(1.3)
0
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 0 6 a 6 L the reference space.
Then u = xt and the two equations in (1.1) reduce to one
g ∂
χ
∂2x
+
= −q̈ .
(1.4)
∂t2
xa ∂a xa
Coupled liquid-vehicle dynamics
3
Assuming nondegeneracy of the label map, xa 6= 0, the wave profile is obtained from
h = χ(a)x−1
a ,
with χ(a) := hxa t=0 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 Lagrangian1 functional
Z t2 Z L Z t2
2
2
2
1
1 χ(a)
1
1
L (x, q) =
dt .
(1.5)
ρχ(a) dadt +
2 (ẋ + q̇) − 2 g x
2 mv q̇ − 2 νq
a
t1
0
t1
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, 10], 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 = 0 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
0 6 y 6 h(x, t)
with 0 6 x 6 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
Dv
1 ∂p
Du 1 ∂p
+
= −q̈ and
+
= −g ,
(2.1)
Dt
ρ ∂x
Dt
ρ ∂y
∂u
∂u
∂u
where g > 0 is the gravitational constant and Du
Dt := ∂t + u ∂x + v ∂y . Conservation of mass in the
Eulerian representation, relative to the body frame, takes the usual form
ux + vy = 0 .
(2.2)
The boundary conditions at the vessel walls are
u=0
at x = 0
and x = L ,
v=0
at y = 0 .
(2.3)
Neglecting surface tension, the boundary conditions at the free surface are
p=0
and ht + uhx = v ,
at y = h(x, t) .
The surface velocity field is defined by
U (x, t) := u(x, h(x, t), t)
1
and
V (x, t) := v(x, h(x, t), t) .
Note that the word “Lagrangian” is used in two distinct ways in this paragraph.
(2.4)
4
H. Alemi Ardakani & T. J. Bridges
The horizontal surface velocity field satisfies the exact equation
h !
Dv hx = −q̈ ,
Ut + U Ux + g +
Dt (2.5)
To verify, differentiate the horizontal surface velocity
h
h
h
h
h
h
h
Du Du Ut + U Ux = ut + uy ht + u(ux + uy hx ) =
+
u
.
(h
+
U
h
−
V
)
=
y
t
x
Dt Dt Hence evaluation of the horizontal component of the momentum equation (2.1) at the free surface
gives
h
1 Ut + U Ux + px = −q̈ .
ρ
But, remarkably,
h
h !
Dv ∂p =ρ g+
hx ,
∂x Dt where
h
Dv Dt (2.6)
is the Lagrangian vertical acceleration at the free surface. This latter identity follows from
integrating the vertical momentum equation in (2.1),
Z
p(x, y, t) = ρg(h − y) +
y
h
Dv
ds ,
Dt
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
ht + U hx = V ,
and recast it into the form
ht + (hU )x = V + hUx .
(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.
2.1 Vehicle motion
If the only force acting on the vehicle is the spring force, then Newton’s second law gives
!
Z LZ h
d
ρu dydx + (mf + mv )q̇ = −νq ,
dt
0
0
(2.8)
where mf is the mass of the fluid:
Z
L
Z
mf =
0
0
h
1
ρ dydx = ρh0 L with h0 =
L
Z
L
h(x, t) dx .
0
RL
The still water level h0 is independent of time because 0 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
Coupled liquid-vehicle dynamics
5
the total fluid momentum and using the horizontal momentum equation in (2.1),
Z LZ h
Z LZ h
d
Du
ρu dydx =
dydx
ρ
dt 0 0
Dt
0
0
Z LZ h
=
(−px − ρq̈) dydx
0
0
h !
Z L
Z h
∂
= −mf q̈ −
p dy − phx dx
∂x 0
0
x=L
Z h
= −mf q̈ −
p dy ,
0
x=0
where Reynold’s transport theorem is used in the first line and the boundary condition p|h = 0 is used
in the last line. Combining this equation with (2.8) gives
Z h
Z h
mv q̈ + νq =
p(L, y, t) dy −
p(0, y, t) dy .
0
0
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
Z LZ h
Z L
Z LZ h
ρudydx =
ρhU dx −
ρyuy dydx .
0
0
0
0
0
Substitute into the governing equation (2.8)
Z
d
dt
!
L
ρhU dx + (mf + mv )q̇
Z
L
Z
h
+ νq =
0
ρyuy dydx ,
0
(2.9)
0
3 Coupled shallow-water sloshing and vehicle motion
The exact equations for (h, U, q) are
Ut + U Ux + ghx + q̈
d
dt
Z
ht + (hU )x
!
L
ρhU dx + (mf + mv )q̇
+ νq
h
Dv hx
= −
Dt = V + hUx ,
Z LZ h
=
ρyuy dydx .
0
0
(3.1)
0
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
Z LZ h
Dv h
≈
0
,
V
+
hU
≈
0
and
ρyuy dydx ≈ 0 .
x
x
Dt 0
0
With these assumptions, the shallow water equations for the fluid are
ht + hUx + U hx
=
0
Ut + U Ux + ghx
= −q̈ ,
(3.2)
coupled to
d
dt
Z
!
L
ρhU dx + (mf + mv )q̇
0
+ νq = 0 .
(3.3)
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
Z L
Z L
d
ρhU dx =
ρ(ht U + hUt ) dx
dt 0
0Z
L = −
ρ (hU 2 )x + ghhx + hq̈) dx
0
Z L
= −mf q̈ −
ρghhx dx .
0
Substitution into (3.3) then gives (1.2).
Cooker [6] shows that the linear problem has two important dimensionless parameters
R=
mv
mf
and G =
νL2
,
4gh0 mf
(3.4)
and they are equally important for the nonlinear problem. The other key dimensionless parameter is
the fluid aspect ratio h0 /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) 7→ (t, x(a, t)) ,
with 0 6 a 6 L ,
t > 0.
Assuming non-degeneracy (xa 6= 0) the derivatives in (1.1) are mapped to
∂
∂
∂
=
− xt x−1
a
∂t
∂t
∂a
and
∂
∂
= x−1
.
a
∂x
∂a
Substitute into (1.1)
−1
−1
ht − xt x−1
a ha + U xa ha + hxa Ua
=
0
−1
−1
Ut − xt x−1
a Ua + U xa Ua + gxa ha = −q̈ .
Setting U = xt and multiplying the first equation by xa simplifies these equations to
xa ht + hxat
xtt + gxa−1 ha
=
0
⇒
d
dt (hxa )
=0
(4.1)
= −q̈ .
Integrating the first equation
hxa = χ(a) ,
where χ(a) := hxa t=0 is determined by the initial data. The second equation of (4.1) then reduces to
g ∂
χ
∂2x
+
= −q̈ .
2
∂t
xa ∂a xa
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
Z L
Z L
Z L
ρhU dx =
ρhU xa da =
ρU χ(a) da .
0
0
0
Therefore Newton’s law for the vessel motion in the LPP setting is
!
Z L
d2
ρx(a, t)χ(a) da + (mv + mf )q(t) + νq(t) = 0 ,
dt2
0
(4.2)
Coupled liquid-vehicle dynamics
7
and the LPP version of Cooker’s equation is
mv q̈ + νq =
a=L
g χ(a)2 .
L x2a a=0
1
2 mf
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
d2
ν
(c.m. + (1 + R)q(t)) = −
q(t) .
dt2
mf
(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
Z t2 Z L
Z t2
d L
(x,
q
+
sb
q
)
=
q
b
(x
+
q
)
ρχ(a)dadt
+
(mv qbt qt − ν qbq) dt ,
t
t
t
ds t1
0
t1
s=0
"
#
Z t2
Z t2
Z L
ν qbqdt .
=
qbt
xt ρχ(a)da + (mf + mv )qt dt −
t1
t1
0
Integrating by parts and using fixed endpoint conditions on the variations qb gives the equation for q
in (4.2).
Now take the first variation of the action integral (1.5) with respect to x,
Z t2 Z L gχ(a)
d x
b
L
(x
+
sb
x
,
q)
=
(x
+
q
)
x
b
+
a ρχ(a)dadt
t
t
t
ds 2x2a
t1
0
s=0
Z t2 Z L gχa
gχxaa
=
−xtt − qtt − 2 +
x
bρχ(a)dadt ,
xa
x3a
t1
0
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 momentum variables
are
w(a, t)
= xt (a, t) + qt (t)
p(t)
=
(mv + mf )qt (t) +
RL
0
(4.4)
xt (a, t)ρχ(a)da .
The Hamiltonian functional is
H(x, q, w, p)
=
RL
1 2
w ρχ da
0 2
+ 12 m1v
R
L
0
−
RL
1
mv p 0
wρχ da +
1 1 2
2 mv p
2
RL
2
wρχ da + 12 νq 2 + 0 12 ρg xχa da .
(4.5)
8
H. Alemi Ardakani & T. J. Bridges
The governing equations in Hamiltonian form are
gχa
x2a
gχ
x3a xaa
−ẇ
=
δH
δx
=
−ṗ
=
δH
δq
= νq
ẋ =
δH
δw
q̇
δH
δp
RL
+ m1v 0 w ρχ(a)da
RL
= m1v p − m1v 0 w ρχ(a)da .
=
−
=w−
1
mv p
(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) = h0 + H(a, t) ,
and linearize the governing equations about the trivial solution. Then χ(a) = h0 and X(a, t) and
H(a, t) satisfy
Xtt − gh0 Xaa = −qtt ,
(5.1)
with boundary conditions X(0, t) = X(L, t) = 0 and
H + h0 X a = 0 ,
(5.2)
and the linearized vehicle motion equation is
!
Z
mf L
d2
X(a, t) da + (mv + mf )q(t) + νq = 0 .
dt2
L 0
(5.3)
Consider solutions that are periodic in time of frequency ω,
X(a, t)
b
= X(a)
sin ωt
h(a, t)
b
sin ωt
= H(a)
q(t)
(5.4)
= qb sin ωt .
Substitution into (5.1)-(5.3) results in the coupled integro-differential system
baa + α2 X
b = −α2 qb ,
X
b + h0 X
ba = 0 ,
H
R
L b
mf
X(a) da + (mf + mv )b
q=
L
0
(5.5)
ν
b,
ω2 q
where
α= √
ω
,
gh0
(5.6)
b
b
and the boundary conditions are X(0)
= X(L)
= 0. The first two equations of (5.5) can be solved for
b
b
qb, X and H,
1
qb = −C
cot 12 αL ,
(5.7)
αh0
where C represents an arbitrary nonzero multiplicative constant,
1
b
X(a)
=C
cot 21 αL(1 − cos αa) − sin αa ,
(5.8)
αh0
and
b
H(a)
= C cos αa − cot 21 αL sin αa .
(5.9)
A key assumption in this derivation is
sin 21 αL 6= 0 .
(5.10)
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
Z L
2
1
b
X(a)
da = C 2
αL cot 21 αL − 1 ,
2
α h0
0
gives
2ρ
C
C
C(s cot s − 1) − mf
cot s = (mv − νω −2 )
cot s ,
α2
αh0
αh0
where
s = 21 αL ,
or since C 6= 0,
g
2 mf αh0 = (−mv ω 2 + ν) cot s .
L
To see that this agrees with Cooker’s result recast in terms of the dimensionless parameters R and G
in (3.4). Then
G
− Rs − tan s ,
(5.11)
∆(s) = 0 , with ∆(s) =
s
which is precisely equation (15) in [6]. Cooker shows that equation (5.11) has a countable number of
solutions, sj with j = 1, 2, . . .. Moreover, for fixed R, G and s > 0 the derivative satisfies ∆0 (s) < 0.
Hence all roots of ∆(s) = 0 are simple. Given a root sj satisfying ∆(sj ) = 0, the natural frequency of
the coupled system is then given by
2p
ωjcoupled =
gh0 sj .
(5.12)
L
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
√
gh0 jπ
f
ωj := 2
, j = 1, 2, . . . .
L
2
The natural frequency of the dry vessel is
r
√
r
ν
gh0 G
ωv =
=2
.
mv
L
R
The natural frequency of the dry vessel plus fluid with the fluid treated as rigid body is
s
√
r
ν
gh0
G
=2
ωf v =
.
mf + mv
L
(1 + R)
Cooker shows that ω1coupled is in fact strictly less than all other natural frequencies.
ω1coupled < ω1f ,
ω1coupled < ωv ,
ω1coupled < ωf v .
(5.13)
The first inequality follows from the fact that s1 < 21 π [6]. Furthermore, ω1coupled → ω1f only in the
limit G → ∞. The third inequality in (5.13) is verified as follows. Use the inequality tan(s1 ) > s1
(since 2s1 < π) and then
r
G
G
⇒ ω1coupled < ωf v .
− Rs1 = tan s1 > s1 ⇒ s1 <
s1
1+R
A similar argument confirms the second inequality in (5.13).
The singularity (5.10) can be expressed in terms of natural frequencies
√
gh0
1
1
sin 2 αL = 0 ⇒ 2 αL = jπ ⇒ ω = 2
jπ .
L
10
H. Alemi Ardakani & T. J. Bridges
But when sj = jπ then tan sj = 0 and so (5.11) reduces to
G
− Rjπ = 0
jπ
or
G
= j 2 π2 ,
R
j = 1, 2, . . . .
Cooker refers to this as resonance, since G = j 2 π 2 R implies that
ωjcoupled = ωv
and ωv = 2ωjf .
The main consequence of the singularity
G
= j 2 π2 ,
R
j = 1, 2, . . . ,
(5.14)
is that the class of periodic solutions (5.4) no longer exists.
6 Numerical algorithm using the Störmer-Verlet method
Discretize the reference space by letting
ai = (i − 1)∆a ,
i = 1, . . . , N + 1 ,
with ∆a =
L
,
N
and let xi (t) := x(ai , t) and wi (t) := w(ai , 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)
q̇ = f (p) ,
for some functions f (·) and g(·) whose precise form will be made clear shortly, and where
p = (p, w1 , . . . , wN +1 )
and
q = (q, x1 , . . . , xN +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 q̇ 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
1
pn+ 2
qn+1
= pn + 21 ∆t g (qn )
1
= qn + ∆t f pn+ 2
pn+1
1
= pn+ 2 + 12 ∆t g qn+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, 10]. 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
χ
g
.
xa xa 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
Z L
gχ(a)
F (x) =
−
ρχ(a)da .
2xa
0
Coupled liquid-vehicle dynamics
11
Discretize using a finite difference formula for the derivative
F (x1 , . . . , xN +1 ) =
N
X
1 gχi ∆a
ρχi ∆a .
2 (xi+1 − xi )
−
i=1
Now take a variation in (x1 , . . . , xN +1 ),
N
X
χ2i−1 ∆a
d g
χ2i ∆a
−
ρξi χi ∆a .
F
(x
+
sξ
,
.
.
.
,
x
+
sξ
)
=
1
1
N +1
N +1
ds s=0
2χi (xi − xi−1 )2
(xi+1 − xi )2
i=1
A semi-discretization of the first equation in (4.6) is then
χ2i−1
χ2i
g∆a
−
,
ẇ =
2χi (xi − xi−1 )2
(xi+1 − xi )2
i = 2, . . . , N .
For the integral of w that appears in (4.6) Simpson’s rule is used
RL
1
1
σ n+ 2 := 0 wn+ 2 ρχda
P N2 −1 n+ 21
P N2
n+ 1
n+ 1
n+ 1
≈ 13 ρ∆a w1 2 χ1 + 2 i=1
w2i 2 χ2i + wN +12 χN +1 .
w2i+1 χ2i+1 + 4 i=1
The above discretization combined with the Störmer-Verlet method then gives the following equations for each time step n 7→ n + 1
1
pn+ 2
n+ 1
wi 2
xn+1
i
= pn −
=
win
+
∆t
n
2 νq ,
g∆t∆a
4χi
2
= xni + ∆twi
∆t
mv p
= qn +
pn+1
= pn+ 2 −
1
n+ 21
= wi
χ2i−1
(xni −xni−1 )
n+ 21
q n+1
win+1
n+ 21
−
∆t
mv σ
∆t
n+1
2 νq
+
g∆t∆a
4χi
+
n+ 21
,
∆t
mv p
−
−
n+ 12
χ2i
2
,
,
i = 2, ..., N
(xni+1 −xni )
∆t
mv σ
n+ 12
i = 2, ..., N ,
(6.1)
,
χ2i−1
2
−xn+1
(xn+1
i
i−1 )
−
χ2i
2
n+1
)
(xn+1
i+1 −xi
,
i = 2, ..., N ,
where
win = w (ai , tn ) ,
pn = p (tn ) ,
xni = x (ai , tn ) ,
q n = q (tn ) ,
tn = n∆t ,
and the boundary conditions are
xn+1
1
=
xn+1
N +1
= L,
n+ 21
0,
n+ 1
w1
= wN +12 =
1
n+ 12
(mv +mf ) p
,
w1n+1
n+1
= wN
+1 =
1
n+1
(mv +mf ) p
.
6.1 Energy conservation
The energy (1.3) of the coupled system, transformed to the LPP setting is
Z L
1
1 χ
1
1
2
E (t) =
(U + q̇) + g
ρχ(a)da + mv q̇ 2 + νq 2 .
2
2 xa
2
2
0
(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 = ai is
Ei =
1
1
ρχi wi2 + ρgχi hi .
2
2
12
H. Alemi Ardakani & T. J. Bridges
Using Simpson’s rule the discrete approximation of the energy in the LPP setting becomes
P N2 −1
P N2
1
1
2
2
E
+
2
E
+
4
E
+
E
E ≈ ∆a
1
2i+1
N +1 + 2 mv q̇ + 2 νq .
i=1
i=1 2i
3
(6.3)
One of the remarkable properties of the Störmer-Verlet algorithm is that the energy error is O(∆t2 )
uniformly in time. This property is proved in [10]. 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(∆tp ), 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 [10] is for ODEs only. However, the important property is that the energy error
is uniform in time, hence long time simulations are possible.
7 Numerical simulations – linear system
The linearized system is
ẇ
= gh0 Xaa ,
ṗ
= −νq ,
Ẋ
= w−
q̇
=
(7.1)
RL
1
+ LR
w da ,
0
R
L
1
1
mv p − LR 0 w da .
1
mv p
This system is also Hamiltonian with functional
RL
RL
1 1 2
0
H(X, q, w, p) = 21 ρh0 0 w2 da − ρh
mv p 0 w da + 2 mv p
2
RL
ρ2 h2 R L
+ 12 mv0 0 w da + 12 νq 2 + 21 ρgh20 0 Xa2 da .
(7.2)
The Hamiltonian functional is the energy of the linear system and in terms of the original variables it
is
Z L
1
1
2
E = 21
(Xt + qt ) + gh0 Xa2 ρh0 da + mv q̇ 2 + νq 2 .
(7.3)
2
2
0
The system (7.1) is discretized using a centred finite-difference formula for Xaa , Simpson’s rule for
the integral, and Störmer-Verlet for time integration. The boundary conditions are
X(0, t) = X(L, t) = 0 ,
w(0, t) = w(L, t) =
1
p(t) .
mv + mf
The first simulation is a comparison with the exact oscillatory solution of §5. For this simulation,
take initial data
X(a, 0) = 0 ,
and q(0) = 0 ,
and for the momenta,
b
w(a, 0) = ωb
q + ω X(a)
and
p(0) = (mv + mf )ωb
q+ω
mf
L
Z
L
b
X(a)
da .
(7.4)
0
Once the parameters are fixed, the frequency ω is obtained by solving ∆(s) = 0 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
g
ν = (mv + mf ) ,
`
where ` is the length of the suspension cable. Based on the range of values in the experiments, the
range of ν is
10.3 < ν < 746.9
kg/sec2 .
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
0.075 < R < 3.935
and 0.32 < G < 73.5 .
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 = 50 kg/sec2 and the frequency ω = ω2coupled . 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
|Xt (a, t) | .
∆a
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 = 170 kg/sec2 and take ω = ω1coupled . 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 = ω2coupled , ν = 150 kg/sec2 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 = ω1coupled , ν = 380 kg/sec2 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 = ω3coupled and G > π 2 R are shown in Figures 9 and 10.
14
H. Alemi Ardakani & T. J. Bridges
Table 1. Input data for the numerical experiments in Figures 2–4. The fluid density is ρ = 1000 kg/m3 ,
the gravitational constant is set at g = 9.81 m/s2 , the tank length is L = 0.525 m, and the still water
level is h0 = 0.09 m. The time step in all cases is ∆t = 10−4 and the number of time steps is typically
106 .
Figure
2 (first row)
2 (second row)
3 and 4
ν (kg/sec2 )
50.0
50.0
170.0
mf (kg)
47.25
47.25
47.25
mv (kg)
6.15
6.15
6.15
C (m)
0.02
0.02
0.03
ω (rad/sec)
10.08113345=ω2coupled
10.08113345=ω2coupled
1.72078580=ω1coupled
h0 /L
0.17143
0.17143
0.17143
R
0.13015873
0.13015873
0.13015873
G
8.25876845
8.25876845
0.28079812
ω1coupled (rad/sec)
0.95730232
0.95730232
1.72078580
ω1f (rad/sec)
5.62271832
5.62271832
5.62271832
ωv (rad/sec)
2.85132972
2.85132972
5.25759222
ωf v (rad/sec)
0.96764124
0.96764124
1.78424230
∆a (m)
0.0026250
0.0001050
0.0001050
tCP U (sec)
60.8598
1296.9010
647.2004
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.
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 = ω2coupled and G < π 2 R.
In Figure 10 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
ωv is equal to ω1coupled and ωv = 2ω1f . However the line G = j 2 π 2 R is not a resonance in the usual
sense since ωv and ω1f 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
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 5–14. The fluid density ρ = 1000 kg/m3 ,
the gravitational constant is set at g = 9.81 m/s2 , and the tank length is L = 0.525 m. In all cases the
space step is ∆a = 1.05 × 10−4 m and time step is ∆t = 10−4 sec, the number of time steps is typically
106 and the CPU time per simulation is typically 20 minutes.
Figure
5 and 6
7 and 8
9 and 10
11 and 12
q̇ (0) = 0.4
13 and 14
q (0) = 0.15
h0 (m)
0.05
0.07
0.07
0.07
0.07
ν (kg/sec2 )
150.0
380.0
380.0
380.0
380.0
mf (kg)
26.25
36.75
36.75
36.75
36.75
mv (kg)
5.80
3.50
3.50
3.50
3.50
C (m)
0.015
0.025
0.025
–
–
ω (rad/sec)
7.51404465
= ω2coupled
2.68446786
= ω1coupled
18.66747146
= ω3coupled
–
–
h0 /L
0.0952
0.1333
0.1333
0.1333
0.1333
R
0.22095238
0.09523809
0.09523809
0.09523809
0.09523809
G
0.80275229
< π2 R
1.03757099
> π2 R
1.03757099
> π2 R
0.93996232
= π2 R
0.93996232
= π2 R
ω1coupled (rad/s)
1.98055989
2.68446786
2.68446786
2.58557248
2.58557248
ω1f (rad/sec)
4.19092679
4.95877146
4.95877146
4.95877146
4.95877146
ωv (rad/sec)
5.08547619
10.41976144
10.41976144
9.91754292
9.91754292
ωf v (rad/sec)
2.16337402
3.07262002
3.07262002
2.92452385
2.92452385
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 = ω1coupled and G > π 2 R.
Figure 8. Snapshots of Eulerian wave profile associated with Figure 7.
conditions are used
h (a, 0)
= h0 ,
w (a, 0)
= q̇ (0) ,
X (a, 0) = 0 ,
q(0) = 0 ,
p (0) = (mv + mf ) q̇ (0) ,
with q̇ (0) 6= 0 specified.
Setting q̇ (0) = 0.40, 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
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 10. Snapshots of Eulerian wave profile associated with Figure 9.
with a quiescent fluid. The complete set of initial conditions is
h (a, 0)
= h0 ,
w (a, 0)
=
0,
X (a, 0) = 0 ,
p (0) = 0 ,
q (0) 6= 0 ,
q̇ (0) = 0 .
Set q (0) = 0.15, 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.
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 11.
8 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 = 130 kg/sec2 , and set the initial velocity of the vehicle at q̇(0) = 0.1.
The other parameter values are listed in the first column of Table 3. The complete set of initial
conditions is
h (a, 0) = h0 , x (a, 0) = a , q(0) = 0 ,
q̇ (0)
=
0.1 ,
w (a, 0) = q̇ (0) ,
p (0) = (mv + mf ) q̇ (0) .
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/sec2 and set the initial position of the vehicle
20
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, 0)
q̇ (0)
= h0 ,
=
0,
x (a, 0) = a ,
w (a, 0) = 0 ,
q(0) = 0.05 ,
p (0) = 0 .
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, 0) = h0 +
− a tan θ0 .
(8.1)
2
Coupled liquid-vehicle dynamics
21
Figure 15. Computed vehicle horizontal motion for long and short time, and energy error when q(0) = 0
and q̇(0) = 0.1 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) := hxa = h (a, 0) = h0 +
t=0
L
− a tan θ0 .
2
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
H. Alemi Ardakani & T. J. Bridges
Table 3. Input data for the numerical experiments in Figures 15–19. The fluid density ρ = 1000 kg/m3 ,
the gravitational constant is set at g = 9.81 m/s2 , and the tank length is L = 0.525 m. The space step
and time step in all cases are ∆a = 1.05 × 10−4 m and ∆t = 10−4 s, the number of time steps is 106
and the CPU time per simulation is typically about 20 minutes.
Figure
15 and 16
17 and 18
19
h0 (m)
0.08
0.06
0.05
ν (kg/sec2 )
130.0
35.0
0.0
mf (kg)
42.0
31.50
26.25
mv (kg)
8.40
11.025
13.125
q̇(0) (m/sec)
0.10
0.0
0.01
q(0) (m)
0.0
0.05
0.0
h0 /L
0.1524
0.1143
0.0952
R
0.20
0.35
0.50
G
0.27176509
0.13007560
0.0
ω1coupled (rad/s)
1.55655499
0.89646318
–
ω1f (rad/sec)
5.30114967
4.59093028
4.19092679
ωv (rad/sec)
3.93397893
1.78174162
0.0
ωf v (rad/sec)
1.60604018
0.90721842
0.0
horizontal restoring force. When ν = 0 the equation for q(t) can be integrated (see Appendix C),
Z L
x(a, t)ρχ(a)da + (mv + mf ) q(t) = c1 t + c0 .
0
Substituting for the initial data gives
1
q(t) = q̇(0) t −
L
Z
L
[x(a, t) − a]
0
χ(a)
da .
h0
(8.2)
The first term gives a uniform drift. With q̇(0) = 0.01 m/sec the drift should amount to about 1 m
after 100 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 10−6 for all time, which is O(∆t3/2 ). When ν = 0 the
first coupled natural frequency, ω1coupled , does not exist since G = 0 and so ∆(s) = −Rs − tan s which
has no solutions in the interval 0 < s < 21 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 = −ν1 q + ν2 q 3 ,
Coupled liquid-vehicle dynamics
23
Figure 17. Computed vehicle horizontal motion for long and short time, and energy error when q(0) = 0.05
and q̇(0) = 0 associated with the first nonlinear simulation.
Figure 18. Snapshots of Eulerian wave profile associated with Figure 17.
where ν1 > 0 and ν2 are constants. If ν2 < 0 the spring is called hard and if ν2 > 0 it is called soft.
With this spring force, Cooker’s equation (1.2) is modified to
mv q̈ + ν1 q − ν2 q 3 = 12 ρg h(L, t)2 − h(0, 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
H. Alemi Ardakani & T. J. Bridges
Figure 19. Computed vehicle horizontal motion and energy error when ν = 0.0 and the initial surface
profile is (8.1) with θ0 = −0.5 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
Z
1 h
u(x, t) =
u(x, y, t) dy ,
h 0
is used then two out of the three equations (3.2)-(3.3) for the coupled problem are exact. Differentiating
!
h Z h
h
h
Z h
Z h
∂
(hu)x =
(ux + vy ) dy −
vy dy = hx u − v = −ht ,
u dy = hx u +
∂x
0
0
0
using mass conservation, the kinematic free surface condition and the bottom boundary condition.
Hence the equation for h is exact
ht + (hu)x = 0 .
Coupled liquid-vehicle dynamics
25
Similarly the equation for the vehicle motion is also exact since (2.8) is in terms of u,
!
Z L
d
ρhudx + (mf + mv )q̇ = −νq ,
dt
0
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
ut + uux + ghx = −q̈ − Rem ,
where the remainder has the form
1∂
Rem =
h ∂x
"Z
h
u2 − u2 +
Z
0
y
h
! #
Dv
ds dy .
Dt
Hence to use u the only assumption required to close the equations is kRemk 1. It is not clear when
this term is small, but sufficient conditions are negligible vertical accelerations and u2 ≈ u2 .
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
Z L
1
2
2
2
2
1
1
1
L (q, h, U ) =
2 ρh(U + q̇) − 2 ρgh + ρφ(ht + (hU )x ) dx + 2 mv q̇ − 2 νq .
0
This Lagrangian functional (for the Eulerian flow field) is of the form
L = KE − P E + constraint ,
where
L
Z
KE =
0
1
2 ρh(U
2
+ q̇) dx +
2
1
2 mv q̇
and P E =
1
2
Z
L
0
ρgh2 dx + 21 νq 2 .
with conservation of mass ht + (hU )x = 0 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
=0 ⇒
∂φ
ht + (hU )x = 0
∂L
=0 ⇒
∂h
1
2 (U
+ q̇)2 − gh − φt − φx U = 0
∂L
= 0 ⇒ h(U + q̇) − φx h = 0
∂u
R
∂L
L
= 0 ⇒ − ddt 0 ρh(U + q̇)dx − mv q̈ − νq = 0 .
∂q
The first equation is mass conservation. The third equation gives
φx = U + q̇ .
Substitute this expression into the second equation,
φt = 12 (U + q̇)2 − gh − (U + q̇)U
⇒
φt + 12 U 2 + gh = 12 q̇ 2 ,
a form of Bernoulli’s equation. Differentiate it
φxt = (U + q̇)Ux − ghx − U Ux − (U + q̇)Ux ,
which simplifies to
Ut + U Ux + ghx = −q̈ ,
26
H. Alemi Ardakani & T. J. Bridges
recovering the momentum equation in (3.2). The variation ∂L /∂q = 0 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
!
Z L
d2
ρxχ(a)da + (mf + mv ) q = 0 ,
dt2
0
or
Z
L
xρχda + (mv + mf ) q = c1 t + c0 ,
(C-1)
0
where c0 and c1 are determined by the initial data. In this case the position of the vehicle can be
expressed in terms of the fluid motion
RL
1
1
0
q (t) = mf c+m
t + mf c+m
− mf +m
xρχda ,
(C-2)
0
v
v
v
with
c1
c0
=
RL
=
RL
0
0
ρU (a, 0) da + (mf + mv ) q̇ (0) ,
x (a, 0) ρχda + (mf + mv ) q (0) .
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
Z LZ h
Z L
1
1
ρx dydx =
x(a, t)ρχ(a) da .
c.m. =
mf 0 0
mf 0
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