Download pdf - Interfacial gravity waves in a two-!uid system

Fluid Dynamics Research 30 (2002) 31–66

Interfacial gravity waves in a two-!uid systemM. La Roccaa ;∗, G. Sciortinoa , M.A. Bonifortib

aDepartment of Civil Engineering Sciences, University RomaTRE, Via Vito Volterra 62,00146 Rome, Italy

bDepartment of Hydraulics-University “La Sapienza”, Via Eudossiana 18, 00184 Rome, Italy

Accepted 5 October 2001

Abstract

In this work a theoretical and experimental investigation is performed on the sloshing of twoimmiscible liquid layers inside of a closed square-section tank. By applying a variational ap-proach to the potential formulation of the !uid motion, a nonlinear dynamical system is derivedapplying the Lagrange equations to the Lagrangian of motion de1ned in terms of suitable gener-alised coordinates. These coordinates are the time depending coe3cients of the modal expansionsadopted for the separation surface of the two !uids and for the velocity potentials of the !uidlayers. Dissipative e4ects are taken into account by considering generalised dissipative forcesderived by a dissipative model extensively treated in the paper.

Numerical integration of the dynamical system furnish solutions which well reproduce the ex-amined experimental con1gurations. c© 2002 Published by The Japan Society of Fluid Mechanicsand Elsevier Science B.V. All rights reserved.

PACS: 47.15.Hg; 47.35.+i; 47.55−t; 02.60.Cb

Keywords: Sloshing; Oscillating tank; Internal waves; Dissipative models; Variational methods; Dynamicalsystems

1. Introduction

The present work consists in a theoretical and experimental study on a strati1ed!uid system. In particular, the interfacial waves generated inside a moving container,with two superimposed liquid layers, are considered. The dynamics of the strati1ed!uid systems it is a very interesting subject both from theoretical and applicative point

∗ Corresponding author.E-mail addresses: [email protected] (M. La Rocca), [email protected]

(G. Sciortino).

0169-5983/02/$22.00 c© 2002 Published by The Japan Society of Fluid Mechanics andElsevier Science B.V. All rights reserved.PII: S0169 -5983(01)00039 -9

32 M. La Rocca et al. / Fluid Dynamics Research 30 (2002) 31–66

of view. From the theoretical point of view, several mathematical models have beenformulated, but many issues need still to be studied in depth. From the applicative pointof view, in many problem of the Applied Fluid Dynamics the hypothesis of strati1ed!uid gives a more realistic description of the examined phenomenon, for instance thedetermination of the dimensioning parameters of the risers of !oating o4shore platform,operating in deep water, needs an accurate knowledge of properties of internal wavesand the loads they may cause.

There is a wide literature concerning interfacial waves, a brief review of which isgiven in the following.

From the experimental point of view, the works of Kalinichenko (1986) andKalinichenko et al. (1991) on a two layer liquid have to be mentioned: the 1rst isfocused on the parametric instability of the interfacial waves, the second on the exper-imental study of the velocity 1eld. A very recent work on the parametric instability ofthe interfacial waves has been performed by Benielli and Sommeria (1998), both ex-perimentally and theoretically. The properties of solitary, interfacial and internal waveshave recently received attention. In particular Grue et al. (1999, 2000) have studiedboth experimentally and theoretically such kind of waves.

From the numerical point of view, there is a huge literary production. It is hereinteresting to mention the works of Michallet and Dias (1999), focused on the solitary,interfacial waves, and of Wright et al. (2000), focused on the numerical study oftwo-dimensional oscillations of a two layer liquid.

From the analytical–numerical point of view, linear or weakly nonlinear analysis areperformed on two layer !uids, expanding the unknown functions, describing kinematicand dynamic quantities, by means of suitable function spaces (Kumar and Tuckermann,1994; King and Mc Cready, 2000). Perturbative techniques have been and are applied tostudy both surface and interfacial waves. In particular, concerning the interfacial waves,the works of Choi and Camassa (1996) and Choi et al. (1996) are to be mentioned, inwhich such perturbative techniques are applied to Euler equation. Among the analyticalapproaches to the interfacial waves dynamics, variational methods with Hamiltonianformulation for the motion equations, have recently obtained great attention. The worksof Benjamin and Bridges (1997), Berning and Rubenchik (1998), Craig and Groves(2000), Ambrosi (2000), are interesting examples of such approach to the study of theinterfacial waves dynamics.

In this paper the interest is focused on the study of the interfacial waves dynamicsinside a moving container, by means of a variational approach with Lagrangian for-mulation for the motion equations (Moiseiev and Rumyantsev, 1968; Whitham, 1974;Miles, 1976). Such an approach has recently been applied to the sloshing of a homo-geneous liquid (La Rocca et al., 2000; Faltinsen et al., 2000), revealing some peculiarfeatures that makes it attractive for the application to the dynamics of interfacial waves.In fact, the variational approach with Lagrangian formulation of the motion equations,can be considered as an energetic formulation of the problem, in which the dissipativee4ects can be easily taken into account by means of the generalised dissipative forces(Goldstein, 1982; Miles, 1976; La Rocca et al., 2000). On the contrary, in the authorsknowledge, it does not result from the literature the introduction of dissipative e4ectsin a variational approach with Hamiltonian formulation of the motion equations. The

M. La Rocca et al. / Fluid Dynamics Research 30 (2002) 31–66 33

introduction of such dissipative e4ects is necessary in order both to perform realis-tic simulations of the interfacial wave dynamics, in particular when large amplitudemotions are considered, and to analyse long duration transients.

The mathematical model is derived assuming that the !uids are inviscid and im-miscible: a potential function can then be de1ned in each layer, in order to de1nethe absolute velocity in the inertial frame of reference. Such assumption is reasonableas dissipative e4ects are con1ned in boundary layers situated near the walls and theseparation surface. Inside the bulk of the two liquid layers, the motion can then beconsidered substantially irrotational (Landau and Lifshitz, 1989; Batchelor, 1987). Asa consequence of the introduction of the velocity potentials, the pressure distributionsin each liquid layer can be calculated by applying the Bernoulli conservation law:such distributions depend on the corresponding velocity potentials and on the motionimposed to the container.

The velocity potentials and the function describing the separation surface — i.e. theunknown quantities describing the dynamics of the examined problem — could bedetermined by solving the mathematical problem constituted by the Laplace equationwithin each liquid layer and suitable linear and nonlinear boundary conditions imposed,respectively, on the container walls and on the separation surface. The main di3cultiesof such mathematical problem are linked to the nonlinear boundary condition, imposedon the separation surface which is an unknown of the problem. Moreover, with thepotential formulation it is not easy to take into account for dissipative e4ects.

Such di3culties can be overcome by adopting a variational formulation. In partic-ular, the problem constituted by the Laplace equation and the boundary conditions isshown to be equivalent to a variational one that consists in 1nding the extrema ofa suitable functional (Tonti, 1984). Such functional is de1ned as the integral over atime interval of the Lagrangian of the motion of the system. The Lagrangian of themotion can be de1ned as the integral of the pressure 1eld over the !uid domain (Luke,1967; La Rocca et al., 2000, Faltinsen et al., 2000). Expanding the separation-surfaceand the velocity potentials in Fourier series with unknown, time-dependent coe3cientsand considering such coe3cients as the generalised coordinates of the motion, it isstraightforward to derive the Lagrange equations for the examined problem. It is usefulto consider such equations because it is known that the solutions of the Lagrange equa-tions coincide with the critical points of the functional. Using the Lagrange equationsit is now possible to account for the damping of the gravity waves in spite of the po-tential formulation. This aim can be achieved introducing generalised dissipative forcesin the Lagrange equations (Goldstein, 1982). Such forces can be derived by a suitabledissipation function (Miles, 1976; La Rocca et al., 2000) which needs the knowledgeof the logarithmic decrement coe3cients of each free-oscillation mode. A consistentpart of this work was devoted to formulate a mathematical model for the accuratecalculation of such logarithmic decrement coe3cients and to their experimental vali-dation. These coe3cients were de1ned as the ratio between the mean-dissipated powerand the mean total mechanical energy of the !uid system (Miles, 1967). The mean dis-sipated power was calculated by integrating the expression of the strain energy over thewhole !uid domain, taking into account for the viscous solutions in the boundary layerregions.


The Lagrange equations are a set of in1nite, nonlinear, 1rst order, di4erential equa-tions for the time dependent coe3cients. Such set of equations is in nonnormal, singularform. The elimination of the singularity by means of a suitable 1rst integral of motionpermits to normalise the di4erential system and then to obtain numerical solutions byusing a 1nite number of equations. In fact, such numerical procedure is able to satisfyin each instant the constraint on the time dependent coe3cients constituted by the 1rstintegral of the motion.

The experimental observations were carried out in a closed, square-section, prismatictank, 1lled with two !uids: coloured water and transparent vaseline oil, separated byan interfacial separation surface. Waves were excited on the interface by an oscillatingrotation imposed to the tank, with varying amplitude and frequency. The motion of theseparation surface was recorded by using laser sensors.

Several experimental con1gurations were examined and compared with the corre-sponding numerical simulations. The good agreement — both qualitative and quantita-tive — between numerical and experimental results con1rms the validity of the presentmodel.

Due to their originality, two aspects of the present work have to be highlighted. First,the adopted de1nition of the Lagrangian it has been widely used for the free surfacewave dynamics (Miles, 1976, 1988; La Rocca et al., 2000; Faltinsen et al., 2000) but,in the authors knowledge, it has been rarely used for the interfacial wave dynamics. Inparticular, although the dynamics of free surface and interfacial waves are similar formany aspects, the mathematical models derived from the aforementioned approach showa deep di4erence. In the dynamics of free surface waves all the nonlinear boundarycondition are evolutionary (La Rocca et al., 2000), while in the dynamics of interfacialwaves the continuity of the normal velocity component on the separation surface leadsto a nonevolutionary condition, making the di4erential system singular.

Second, the mathematical model is de1ned taking into account the dissipative ef-fects, introducing suitable generalised dissipative forces. To this aim, a modi1ed Miles(1967) approach has been followed. In the present approach the dependence of thevelocity in boundary layer regions on the coordinate normal to the boundary layer istaken into account, due to a di4erent de1nition of the mean dissipation rate. Such anapproach takes into account the dissipation e4ects due to the presence of the walls,which have been ignored in previous works (La Rocca et al., 2000). Finally, the in-troduction of the dissipative e4ects in the mathematical model has to be highlighted asan advantage of the Lagrangian formulation of the problem respect to a Hamiltonianone.

2. Problem formulation

A closed, square section prismatic container of height H and side B is completely1lled with two immiscible !uids whose densities are �1 and �2 (�1 ¿�2). At rest, thelayer of the !uid, whose density is �1, reaches the level H1; measured with respect tothe bottom of the tank, while the thickness of the second liquid layer is H2 (Fig. 1).


Fig. 1. The experimental setup.

Let the frame of reference Oxyz be attached to the container, with i; j; k unit vectorsof x; y; z axes, respectively, as de1ned in Fig. 1. The level z = 0 coincides with theseparation surface between the two layers at rest.

The container can rotate at an angle around an axis R parallel to the y-axis(Fig. 1), where = (t) is a given function of the time t.

The Euler equation, considered in each !uid layer with the apparent forces, is

DujDt

+ 2W ∧ uj + W′ ∧ r + W ∧ (W ∧ r) = − 1�j

∇pj + f ; j = 1; 2; (1)

where f=g∇[(x−B=2) sin()−z cos()]), W=X′(t)j (the prime denotes di4erentiationwith respect to time) and r = (x − B=2)i + (y − B=2)j + (R + H1 + z)k.

Assuming the motion to be irrotational in the inertial frame of reference, the !uidvelocity uj, relative to the tank, can be expressed in each layer as

uj(x; y; z; t) = ∇’j(x; y; z; t) −W ∧ r; j = 1; 2; (2)

where the term ∇’j represents the absolute velocity 1eld of the jth layer in the inertialframe of reference. The potential functions ’j satisfy Laplace’s equation in the domainsDfj − @Dfj:

∇2’j = 0; j = 1; 2; (3)


de1ned by

Df1 ≡ {(x; y; z) | 06 x6B; 06y6B;−H16 z6 �};Df2 ≡ {(x; y; z) | 06 x6B; 06y6B; �6 z6H2};

where z = �(x; y; t) is the elevation of the separation surface over the plane z = 0.On the container’s walls, the normal component of the relative velocity uj has to be

zero. This gives the following linear boundary condition:

∇’j · n = W ∧ r · n: (4)

Substituting (2) in (1), the following expression for the pressure pj is obtained forthe jth layer:

pj

�j=−

(@’j

@t+

12∇’j ·∇’j−∇’j ·W ∧ r−g[(x−B=2) sin()− z cos()]

):

(5)

On the separation surface z = �(x; y; t) the following boundary condition, in accor-dance with the potential formulation, must be imposed:

(p1 − p2)n = −�(

1R1

+1R2

)n; (6)

where � is the surface tension at the separation surface between the two liquids, n isthe unit normal to the interface and R1; R2 are the radii of curvature of the interfacein any two orthogonal planes containing the normal n (n is considered positive whenthe centre of curvature lies on the side of the interface to which n points).

Surface tension e4ects can be neglected if ��(�1 − �2)gB2 (Christodoulides andDias, 1994). Considering the dimensions of the adopted experimental setup and thevalues of the densities, surface tension e4ects will be neglected in this work.

The kinematic conditions on the separation surface z = �(x; y; t) seen by each layer,are expressed by the following relations:

@�@t

+@�@x

[@’1

@x− ′(R + H1 + �)

]+

@�@y

@’1

@y− @’1

@z− ′

(x − B

2

)∣∣∣∣z=�

= 0;

@�@t

+@�@x

[@’2

@x− ′(R + H1 + �)

]+

@�@y

@’2

@y− @’2

@z− ′

(x − B

2

)∣∣∣∣z=�

= 0:

(7)

The main di3culty of the problem de1ned by Eqs. (3) with boundary conditions(4), (6) and (7), is the nonlinearity of the boundary conditions (6) and (7), in whichp1; p2 are expressed by (5).

3. Variational approach

Physical problems are usually formulated by means of di4erential equations. Never-theless, in some problems it can be very useful to transform such kind of formulationin an equivalent variational formulation.


As it is well known, the crucial point in a variational formulation is the de1nitionof a suitable functional F whose extrema coincide with the solution of the examinedproblem. In this work, the following de1nition for F is adopted (La Rocca et al., 2000;Faltinsen et al., 2000):

F ≡∫ t1

t0L dt;

L ≡⟨∫ �

−H1

p1 dz⟩

+⟨∫ H2

�p2 dz

⟩;

〈·〉 ≡∫ B

0

∫ B

0dx dy; (8)

where L is the Lagrangian of the motion (Whitham, 1974). Substituting expression (5)for the pressure 1elds in (8), it becomes clear that the functional F depends on theunknown functions ’1; ’2; �, i.e. F = F(’1; ’2; �):

It is possible to show that the boundary value problem given by (3), with the bound-ary conditions (4), (6), (7) and the assigned initial conditions for t = t0 is equivalentto impose that the 1rst variation of F , calculated with respect to ’1; ’2; �, is equal tozero (Luke, 1967; Tonti, 1984; La Rocca et al., 2000):

�F = 0 (9)

with �’1(x; y; z; ti) = 0; �’2(x; y; z; ti) = 0; ��(x; y; ti) = 0; i = 0; 1:In other words, the solution of the abovementioned boundary value problem

(’1; ’2; �) makes the functional stationary.The following modal expansions for the functions ’1; ’2; � are now introduced:

’1(x; y; z; t) = ′(t)’p1(x; z) +∞∑

i(n;m)=1

�1; i(n;m)(x; y; z; t);

’2(x; y; z; t) = ′(t)’p2(x; z) +∞∑

i(n;m)=1

�2; i(n;m)(x; y; z; t);

�(x; y; t) =∞∑

i(n;m)=1

Qi(n;m)(t) cos( nx) cos(�my); (10)

where �1; i(n;m)(x; y; z; t); �2; i(n;m)(x; y; z; t) are de1ned in the following:

�1; i(n;m)(x; y; z; t) = A1; i(n;m)(t)cosh["i(n;m)(H1 + z)]

cosh["i(n;m)H1]cos( nx) cos(�my);

�2; i(n;m)(x; y; z; t) = A2; i(n;m)(t)cosh["i(n;m)(z − H2)]

cosh["i(n;m)H2]cos( nx) cos(�my): (11)


The de1nitions of the unknown symbols and of ’p1(x; z); ’p2(x; z) are given in Ap-pendix A. Moreover, a single index notation i = i(n; m) will be used, which turns outto be very useful to produce a compact matrix formulation for the motion equations.

As it is possible to see in (10) and (11), the potentials ’1; ’2 are decomposed intothe sum of particular solutions ’p1; ’p2 and homogeneous solutions �1; i(n;m); �2; i(n;m)

of the Laplace equations (3). The basis of functions used for the expansions (11) iscomplete for the examined problem. In fact, the basis {cos( nx) cos(�my)} is com-plete in the square-integrable space functions, de1ned in the domain: {(x; y)∈ (0; B)×(0; B)} (Kolmogorov and Fomin, 1980), while the functions (cosh["i(n;m)(z ± H1;2)])=(cosh["i(n;m)H1;2]) derive from the harmonicity of the potentials �1;2i(n;m). The particularsolutions ’p1 and ’p2 satisfy the linear boundary condition (4), while the homoge-neous solutions �1; i(n;m) and �2; i(n;m) satisfy the homogeneous linear boundary condi-tion ∇�1; i(n;m) · n = 0; ∇�2; i(n;m) · n = 0 on the rigid walls of the container for any t(La Rocca et al., 2000). The functions �1; i(n;m) and �2; i(n;m) are completely de1nedwhen the time-dependent coe3cients A1; i(n;m)(t); A2; i(n;m)(t) are determined. Such coef-1cients are obtained by requiring the ful1llment of the nonlinear boundary conditions(6) and (7).

Substituting expansions (10) and (11) into (8), the Lagrangian function is obtainedas a function of the vectors:

A1(t) ≡ {A1; i(n;m)(t)}; A2(t) ≡ {A2; i(n;m)(t)}; Q(t) ≡ {Qi(n;m)(t)}:

As previously mentioned, the introduction of the Lagrangian function permits to imposeeasily the stationarity condition (9). In fact, the 1rst variation of the functional F isequal to zero if the vectors A1(t);A2(t);Q(t) satisfy the Lagrange equations (Whitham,1974; Moiseiev and Rumyantsev, 1968; Goldstein, 1982):

ddt

@L@A′

1− @L

@A1= 0;

ddt

@L@A′

2− @L

@A2= 0;

ddt

@L@Q′ −

@L@Q

= 0:

In order to take into account the damping e4ects of gravity waves, we add to theseequations dissipative terms in the following way:

ddt

@L@A′

1− @L

@A1=

@G@A′

1;

ddt

@L@A′

2− @L

@A2=

@G@A′

2;

ddt

@L@Q′ −

@L@Q

=@G@Q′ ; (12)


where G is a suitable dissipation function (Ceschia and Nabergoj, 1978; Goldstein,1982; La Rocca et al., 2000). The Lagrangian L and the dissipation function G havethe following structure:

L≡ �1

∑i(n;m)

〈M1; i(n;m)(Q; x; y; t)〉dA1; i(n;m)

dt+ 〈N1(A1;Q; x; y; t)〉

−�2

∑i(n;m)

〈M2; i(n;m)(Q; x; y; t)〉dA2; i(n;m)

dt+ 〈N2(A2;Q; x; y; t)〉

G ≡ (�1 − �2)gB2

∑i

&i(n;m)

!2i(n;m)

(dQi(n;m)=dt)2; (13)

where !i(n;m) and &i(n;m) denote, respectively, the linear natural frequency and the log-arithmic decrement of the i(n; m)th mode, M1; i(n;m); M2; i(n;m); N1; N2 are nonlinear func-tions of the variables A1;A2;Q; x; y; t. In Appendix A the quantities !i(n;m) and thefunctions M1; i(n;m); M2; i(n;m); N1; N2 are de1ned, while the calculation of the coe3cients&i(n;m) will be illustrated in the next section.

4. Calculation of the damping coe!cients

In this work, the damping of waves generated on the separation surface is taken intoaccount by introducing the dissipation function G de1ned in (13). The crucial point inthe de1nition of G consists in the calculation of the logarithmic decrement &i(n;m) ofthe corresponding i(n; m)th mode.

The coe3cients &i(n;m) can be de1ned as (Miles, 1967; Henderson and Miles, 1994;Landau and Lifshitz, 1989):

&i(n;m) =Di(n;m)

2Ei(n;m); (14)

where Di(n;m) and Ei(n;m) are, respectively, the mean dissipation rate and the mean en-ergy (twice the mean kinetic energy for a free oscillation) of the corresponding i(n; m)thmode. For each mode the mean dissipation rate is the sum of the following contribu-tions: viscous dissipation at the rigid boundary of the container, viscous dissipation inthe interior of both !uids, viscous dissipation at the separation surface and dissipationdue to the capillary hysteresis at the contact line (Henderson and Miles, 1994).

The order of magnitude of the 1rst two contributions was found to be respectivelyproportional to "1=2

j ; "j; being "j the kinematic viscosity of the jth !uid layer (j=1; 2):The contribution to Di(n;m); due to the viscous dissipation at the separation surface wasfound proportional to a weighted average of "1=2

1 and "1=22 : It is interesting to note the

di4erence with the case of a single !uid with a free surface for which the contribution to


Di(n;m) due to the viscous dissipation at the free surface is proportional to "3=2j (Mei

and Liu, 1973). The capillary hysteresis is con1ned to a small layer that moves withthe contact line between the !uids and the rigid wall and has a lateral thickness of theorder of capillary length: lc ≡ (�=�g)1=2 (Henderson and Miles, 1994). The contributionof the capillary hysteresis becomes important, in comparison of the other contributions,when a characteristic length of the motion l is such that lc=l 1. In this work, beingl = B and lc=l�1, this contribution can be neglected.

The mean dissipation rate of the i(n; m)th mode due to the viscous stresses can bede1ned as

Di(n;m) ≡ !i(n;m)

2*

∫ 2*=!i

0

(∫V1

2,1D1: D1 dV1 +∫V2

2,2D2: D2 dV2

)dt; (15)

i.e. as the work done by the viscous stresses in the whole !uid domain. Such !uiddomain is the sum of the domains of each !uid layer V1; V2 which coincide withDf1; Df2 assuming � = 0 in the linear regime. In the de1nition (15) D1;D2 are thesymmetrical parts of the tensor ∇uj: In this section uj is the velocity 1eld due tothe free oscillations in the jth !uid layer which can be decomposed into a sum of arotational part uRj associated to the jth !uid layer (sensibly di4erent from zero onlyin the boundary layers of the !uid domain) and in an irrotational part ∇�j

uj = uRj + ∇�j; (16)

where

�j ≡ Re(’jeI!i(n;m)t); ’j ≡ Ajcosh["i(n;m)(z + (−1)j−1Hj)]

cosh["i(n;m)Hj]cos( nx) cos(�my);

Aj = const:

First, we consider the problem of determining the uj velocity in the boundary layergenerated by the free oscillations of the system. As a consequence, considering thelinearised Navier–Stokes equation without apparent forces for the jth !uid layer andsubstituting in this latter decomposition (16), the following equation is obtained:

∇(@’j

@t+

pj

�j+ gz

)+

@uRj@t

= "j∇2uRj: (17)

Imposing that ∇(@’j=@t + pj=�j + gz) = 0, the Stokes equation for the rotationalcontribution to the velocity 1eld in the jth !uid layer is obtained:

@uRj@t

= "j∇2uRj: (18)

Eq. (18) has to be solved in correspondence of the rigid plane walls of the containerand of the separation surface, with di4erent boundary conditions. In particular, consid-ering a system of Cartesian coordinates, x1; x2; x3, with the plane of the rigid wall asthe plane x3 = 0, Eq. (18) and the relative boundary conditions are then expressed by


the following:

@uRj

@t= "j

@2uRj

@x23

;@vRj@t

= "j@2vRj@x2

3;

uRj = 0; vRj = 0; x3 → ∞;

uRj = − @’j

@x1

∣∣∣∣x3=0

eI!i(n;m)t ; vRj = − @�j

@x2

∣∣∣∣x3=0

eI!i(n;m)t ; x3 = 0; (19)

where uRj; vRj are the velocity components parallel, respectively, to x1 and x2: In thissection the convention is adopted consisting in taking into account only the real partof the complex expressions. The solutions of (19) are expressed by

uRj = − @�j

@x1

∣∣∣∣x3=0

e−√

!i(n;m)

2"jx3eI(!i(n;m)t−

√!i(n;m)

2"jx3)

;

vRj = − @�j

@x2

∣∣∣∣x3=0

e−√

!i(n;m)

2"jx3eI(!i(n;m)t−

√!i(n;m)

2"jx3)

; (20)

where√

2"j=!i(n;m) is assumed as the boundary layer thickness, due to the oscillationof the i(n; m)th mode (Landau and Lifshitz, 1989; Batchelor, 1987).

Assuming x3 ≡ z, because of the linearity of the present approach, Eq. (18) is con-sidered in each layer in the neighbourhood of the separation surface with the followingboundary conditions:

uR1 = 0; vR1 = 0; z → −∞;

uR2 = 0; vR2 = 0; z → ∞;

uR1 +@’1

@xeI!i(n;m)t = uR2 +

@’2

@xeI!i(n;m)t ; z = 0;

vR1 +@’1

@yeI!i(n;m)t = vR2 +

@’2

@yeI!i(n;m)t ; z = 0;

,1@@z

(uR1 +

@’1

@xeI!i(n;m)t

)= ,2

@@z

(uR2 +

@’2

@xeI!i(n;m)t

); z = 0;

,1@@z

(vR1 +

@’1

@yeI!i(n;m)t

)= ,2

@@z

(vR2 +

@’2

@yeI!i(n;m)t

); z = 0; (21)

where, as usual, z is the coordinate perpendicular to the separation surface. Theseconditions assure the continuity of the velocity and of the tangential stress through thesurface separations. The analytical expressions for uRj; vRj are omitted for the sake ofsimplicity.


The calculation of the mean dissipation rate of the i(n; m)th mode can then beperformed by applying (15), where Dj is de1ned as

Dj ≡ 12 (grad(uj) + grad(uj)T): (22)

The integral (15) must be calculated considering separately the contributions due theviscous stresses in the boundary layers near the walls, in the boundary layer near theseparation surface and in the interior of both the !uid layers, where the motion isirrotational and then the velocity 1eld is expressed by uj = ∇’j.

The mean total energy of the i(n; m)th mode is de1ned as twice the mean kineticenergy of the i(n; m)th mode (Landau and Lifshitz, 1989):

Ei(n;m) =!i(n;m)

2*

∫ 2*=!i

0

(�1

∫V1

u1 · u1 dV1 + �2

∫V2

u2 · u2 dV2

)dt: (23)

The logarithmic decrement of the i(n; m) mode can be calculated applying de1ni-tion (14). It is to be observed that such quantity depends on the ratio A1=A2 (see thede1nition of ’j), which is equal to −Tanh["i(n;m)H2]=Tanh["i(n;m)H1] by virtue of thelinearised kinematic boundary conditions (7) on the separation surface. Moreover, thelogarithmic decrement of the i(n; m) mode depends on �1 − �2, by means of the res-onance modal frequency !i(n;m); in a very complicated way, but the whole expressionis omitted, for the sake of simplicity.

Finally it is interesting to note that the present approach, in calculating the loga-rithmic decrement of the i(n; m)th re!ects the approach of Miles (1967), but the maindi4erence with such an approach consists in the de1nition of the boundary layer con-tribution to the modal mean dissipation rate. In the present approach such quantity isde1ned as the integral of the strain energy over the boundary layer zones, while theapproach of Miles de1nes the modal mean dissipation rate due to the boundary layers,as the work done by the viscous stresses on the boundary layer’s edge. In the authorsopinion, the present approach has the merit of highlighting the dependence of the ve-locity and, as a consequence, of the strain energy (15) on the coordinate perpendicularto the boundary layer.

5. The mathematical structure of the Lagrange equations

Applying Eqs. (12) to the Lagrangian L the following dynamical system is obtained: %1〈MT1 〉 −%2〈MT

2 〉 0

0 0 %1〈M1〉0 0 %2〈M2〉

A1

A2

Q

=

%2〈@N2=@Q〉 − %1〈@N1=@Q〉 −D

%1〈@N1=@A1〉%2〈@N2=@A2〉

(24)


where for j = 1; 2:

Mj ≡((

@Mj; i(n;m)

@Qk(n;m)

));

D ≡(

@G

@Qi(n;m)

Qi(n;m)

):

The block matrix at l.h.s. of (24) is singular for any squared matrix Mj and it fol-lows that it is impossible to put the dynamical system (24) into normal form. Sucha singularity re!ects the fact that the boundary conditions (7) can be reduced to anonevolutionary condition, by eliminating @�=@t from them. Such nonevolutionary con-dition constitutes a constraint that the unknowns have to satisfy for any instant. Thisdi3culty could be overcome by adopting a Hamiltonian formulation for the motionequations (24), with canonical variables:

1 ≡ �(x; y; t); ≡ �1’1(x; y; �(x; y; t); t) − �2’2(x; y; �(x; y; t); t):

(Berning and Rubenchik, 1998; Ambrosi, 2000; Craig and Groves, 2000). The useof such Hamiltonian canonical variables is, of course, the most elegant way of over-coming the di3culty constituted by the nonevolutionary condition. Nevertheless, suchan approach has some operative limitations. First, following a purely variational Hamil-tonian formulation, it is (in the authors knowledge) quite di3cult to take into accountthe dissipative e4ects, which, on the contrary, are easily introduced in a variational La-grangian formulation by means of the generalised dissipative forces (Goldstein, 1982).Second, such Hamiltonian canonical variables are de1ned on the separation surface andfrom their knowledge it is not possible to go back in a simple way to the value of thesingle velocity potential in each !uid layer, whose knowledge permits the representationof the kinematic and pressure 1elds. For these reasons, in this paper the Lagrangianformulation for the motion equations is adopted and the di3culty, previously described,is overcome in the following way. Eliminating Q from the second and third block ofEqs. (24), the algebraic vector equation

3(A1;A2;Q; t) = 0

is obtained. 3(A1;A2;Q; t) is de1ned as

3(A1;A2;Q; t) ≡ 〈M1〉−1〈@N1=@A1〉 − 〈M2〉−1〈@N2=@A2〉 = 0; (25)

3(A1;A2;Q; t) is then a 1rst integral of the !uid motion. Di4erentiating (25) withrespect to the time t, the following equation is obtained:

d3dt

=@3@A1

A1 +@3@A2

A2 +@3@Q

Q +@3@t

= 0; (26)


which can take the place of the third block of Eqs. (24): %1〈MT1 〉 −%2〈MT

2 〉 0

0 0 %1〈M1〉@3@A1

@3@A2

@3@Q

A1

A2

Q

=

%2〈@N2=@Q〉 − %1〈@N1=@Q〉 −D

%1〈@N1=@A1〉−@3=@t

: (27)

It is possible to verify that the dynamical system (27) is not singular and normalisable,in order to be numerically integrated.

6. The numerical procedure

The numerical integration was performed putting into normal form the dynamicalsystem (27). In particular, for a given law of the forcing oscillation = (t), assignedinitial conditions at time t=t0, a fourth order Runge–Kutta method was applied togetherwith a particular numerical technique explained in the following.

The main di3culties in performing the numerical integration of the dynamical sys-tem (27) are of two kinds. First, at each time step of the numerical integration thecoe3cients of system (27) require to be numerically evaluated, because they cannotbe analytically calculated applying the operator 〈·〉 due to their complex mathematicalstructure. It has to be noted that this procedure draws away from the usual methodwhich expands in Taylor series of �(x; y; t) the coe3cients of the system (27) and thenanalytically integrates such expansions in the domain 06 x6B; 06y6B: With thislast method, 1ctitious nonlinearities are introduced which can generate not physicallyreliable solutions (La Rocca et al., 2000). Such problem can be easily overcome byusing an e3cient technique of numerical evaluation for the integral 〈·〉 ≡ ∫ B

0

∫ B0 dx dy.

The second di3culty which occurs in the numerical integration of (27) is due tothe existence of relation (25), which implies that initial conditions at a 1xed instantt = t0 cannot be assigned in an arbitrary way, but have to satisfy 3(A1(t0);A2(t0);Q(t0); t0) = 0.

As can be easily seen, condition 3(A1;A2;Q; t) = 0 imposes that the velocity com-ponent along the normal to the separation surface is the same for each layer, assuringthe continuous contact of the two !uids along the separation line during the motion.Such a problem is more delicate and will be described in more details. First, it can beshown applying the Lagrangian equations to the Lagrangian L and taking into accountthe de1nitions in Appendix A, that the functional dependence of 3 on A1;A2 is linear,while the dependence on Q is not. Then, supposing that the law of motion (t) is zerofor t6 t0 it follows that Q(t0) = 0 (the separation surface is horizontal at rest). Theinitial conditions can then be chosen among those satisfying 3(A1(t0);A2(t0); 0; t0)=0:In particular, assigning A1(t0) = 0 it is possible to solve the previous linear equation


in the unknown A2(t0): Once the initial conditions are known, the integration of thedynamical system (27) can start.

Nevertheless, the numerical procedure can be a4ected by another problem. In fact,from a strictly mathematical point of view, if initial conditions satisfy the equation3 = 0, the solution of the di4erential equation d3=dt = 0 assures that 3 remains zerofor any time t. Instead, several tests performed by numerical integration of system (27)showed that condition 3=0 is satis1ed more and more roughly with the increasing ofthe simulation time. The violation of condition (25) unbalances the dynamical systemin such a way that numerical results become completely unreliable and a suitableprocedure has to be followed in order to obtain numerical solutions which, for anyinstant t; satisfy condition (25). To pursue this aim, let F be de1ned by the followingformula:

F=∑

i

[(A1; i(n;m) − A1; i(n;m))2 + (A2; i(n;m) − A2; i(n;m))2

+ (Qi(n;m) − Qi(n;m))2 + 4i(n;m)3i(n;m)] (28)

where A1; i(n;m); A2; i(n;m); Qi(n;m) are the values predicted by the application of the fourthRunge–Kutta method (at a given time step) to the dynamical system (27) and 3i(n;m)

is the i(n; m)th component of the vector 3 while 4i(n;m) is the i(n; m)th Lagrangemultiplier. Now, we minimize F with respect to A1; i(n;m); A2; i(n;m); Qi(n;m) and 4i(n;m),that is equivalent to minimize the square deviation

N∑i=1

[(A1; i(n;m) − A1; i(n;m))2 + (A2; i(n;m) − A2; i(n;m))2 + (Qi(n;m) − Qi(n;m))2]; (29)

subjected to the functional constraint (25). The value A1; i(n;m); A2; i(n;m); Qi(n;m) obtainedby solving the algebraic equations:

@F@A1; i(n;m)

= 0;

@F@A2; i(n;m)

= 0;

@F@Qi(n;m)

= 0;

@F@4i(n;m)

= 0; (30)

can be considered as corrected values respect to the predicted values A1; i(n;m);A2; i(n;m); Qi(n;m): In other words, this technique allows to satisfy rigorously the condition3 = 0 for each time step, choosing the values A1; i(n;m); A2; i(n;m); Qi(n;m) among thosewhich are closest (in the least-squares sense) to those predicted by the Runge–Kuttamethod.


7. The experimental setup

The experimental setup consists in a closed, square section tank (0:5 m × 0:5 m ×0:14 m), made of plexiglas and put on a massive support anchored to the !oor in orderto reduce the induced vibrations.

The tank was 1lled with coloured water (�1 1000 kg m−3) up to a level H1 =0:08 m with respect to the bottom, measured in horizontal position. A layer of trans-parent vaseline oil (�2 850 kg m−3); of thickness H2 = 0:06 m; was put over thewater.

The tank can rotate around an axis R parallel to the y axis (Fig. 1), due to a crankmechanism, moved by an electric engine that rotates with a constant angular velocity5. It is possible to vary both the amplitude 0 and the frequency f of the oscillatingrotation in the range:

06 06 0:14 rad; 06f6 1 Hz: (31)

The law of motion is not purely sinusoidal and can be approximated by the followingexpression (La Rocca et al., 2000):

(t) = 0 sin(2*ft) − 0:3120 (1 + cos(4*ft))

+ 30(0:16 cos(2*ft) − 0:16 cos(6*ft)

+0:13 sin(2*ft) − 0:004 sin(6*ft)) + O(40): (32)

For 0�1; the (32) gives: (t) 0 sin(2*ft).

8. Experimental and numerical results

The experimental results presented in this work are concerned both with the eval-uation of the logarithmic decrement coe3cients &i(n;m) of the considered modes andwith the realisation of several motion conditions. Time histories of the elevation of theseparation surface z = �(x; y; t) at suitable measurement points and in di4erent exper-imental conditions were acquired. Moreover, pictures of the separation surface, takenby 1lming the evolution of the process with a digital videocamera, were obtained.The numerical results were carried out in order to reproduce the motion conditionsexperimentally analysed, so that it is possible to validate our mathematical model bycomparing experimental and numerical results.

8.1. Experimental evaluation of the logarithmic decrement coe8cients &i(n;m)

The mathematical model is completely de1ned as soon as the logarithmic decrementcoe3cients &i(n;m) of the considered modes are assigned. In this work such coe3cientsare calculated by mean of formula (14). It is suitable to introduce an experimentalprocedure able to evaluate such coe3cients, in order to compare the &i(n;m) given byformula (14) with those experimentally evaluated. To this extent, let the separation


surface at the point x; y be represented by the following truncated series:

�(x; y; t) =N∑

i(n;m)=1

Qi(n;m)(t) cos( nx) cos(�my); (33)

where N is the total number of the considered modes. Then, it is evident that if Nmeasurements of the time history of the separation surface �(x; y; t) can be contempo-raneously performed in N di4erent measurement points x; y, an algebraic, linear systemof equations in the unknowns Qi(n;m)(t) can be derived from (33). If the separationsurface performs free oscillations, it is then possible, considering expression (33), toseparate the contribution of each mode in the elevation of the separation surface andthen to determine the experimental decay of the mode i(n; m). Our experimental setuppermitted to simultaneously apply three laser sensors, so (33) can be used as an alge-braic, linear system of three equations in three unknowns. As experimentally detected,a 2D representation for the separation surface (m = 0) can be adopted. By using the1rst three modes the (33) calculated in correspondence of the points x1 = 0; x2 = B=2;x3 = B=6; on the plane y = B=2 gives the following system:

Q1 + Q2 + Q3 = �(

0;B2; t)

;

Q2 = −�(B2;B2; t)

;

Q1 cos(*

6

)+ Q2 cos

(*3

)= �

(B6;B2; t)

; (34)

where i(j; 0) ≡ j.The choice of the measurement points is linked to the choice of the modes i(n; m):

in particular the point (B=2; B=2) is a nodal point for the odd modes, while the point(B=6; B=2) is a nodal point for the mode (3; 0). The free oscillations of the 1rst threemodes Q1; Q2; Q3 are then given by

Q1 =(�(B=2; B=2; t) + 2�(B=6; B=2; t))√

3;

Q2 = −�(B2;B2; t)

;

Q3 =1√3

(√3�(

0;B2; t)

+ (√

3 − 1)�(B2;B2; t)− 2�

(B6;B2; t))

: (35)

Once Q1; Q2; Q3 are known, a least square approximation technique can be applied inorder to evaluate the coe3cients &j: In particular, the minimization of the quantity:

Ndata∑k=1

[Qj(kSt) − Aje−&j(kSt) sin(!j(kSt) − �j)]2; (36)

where St is the experimental time interval of sampling with respect to Aj; &j; �j gives asystem of algebraic, transcendental equations for the unknowns Aj; &j; �j. The solution


Table 1

& Numerical Experimental

&1 0.038 0.040&2 0.052 0.058&3 0.066 0.069

of such a system gives the experimental values of the coe3cients &j. The experimentalfree oscillations were realised putting the tank in oscillation with frequency f andamplitude 0. The frequency f is closed to the value of the resonance frequency ofthe 1rst mode because in correspondence of such value the amplitude of the sloshingis large enough in order to excite the 1rst three modes. After stationarity was reached,the electric engine was suddenly switched o4, when the angular position of the tank is(t)=0, generating a free oscillation of the separation surface decaying with the time.

In Table 1 numerical and experimental values of the 1rst three values of &j arepresented:

In Fig. 2 the comparison between the numerical and experimental free-oscillations ofQ1; Q2; Q3 is shown. The numerical free oscillations were obtained by putting 0=0 andby integrating system (27) starting from nonzero initial values. The numerical valuesof the coe3cients &j (j = 1; 2; 3), shown in Table 1, were used for the numericalsimulations.

The analytical expression for the coe3cient &j is very complicate. In particular &j =&j(�1; �2; "1; "2; H1; H2; B; g): Then, applying the Buckingham theorem it is possible toexpress &j as a function of the following nondimensional parameters: 61 = &j

√B=g;

62 = �2=�1; 63 = "1=√

gB3; 64 = "2=√

gB3; 65 = H1=B; 66 = H2=B. In Fig. 3a 61 isplotted versus 64 for the 1rst ten modes (j=1; : : : ; 10), being the other nondimensionalparameters determined by the examined experimental condition and kept constant. Thevertical dotted line characterises the nondimensional kinematic viscosity of the vaselineoil used in the experimental setup. It is interesting to observe that the value of 61

increases, increasing the number of the mode j. Moreover for j¿ 6 the curvature ofthe plotted lines changes its sign increasing 64.

In Fig. 3b 61 is plotted versus 62 for the 1rst 10 modes (j = 1; : : : ; 10), with63; : : : ; 66 kept constant. 62 varies in the range 0662 ¡ 1: It is possible to see that61 become singular for 62 = 1. In the 1gure, this fact is evident for j¿ 6. Suchbehaviour is due to the fact that the limit case 62 = 1 corresponds to a containercompletely 1lled with !uid of density � = �1 = �2: As a consequence, the naturalfrequency !j of the mode (j; 0) tends to zero because the system loses the possibilityof performing free oscillations.

8.2. Validation of the mathematical model

The numerical results presented in this subsection were obtained in order to simulatethe considered experimental con1gurations. The comparison between numerical andexperimental results allowed us to validate our mathematical model.


Fig. 2. Free oscillations of the modes (1; 0); (2; 0) and (3,0).


Fig. 3. (a and b) Plots of &j=(g=B)1=2, respectively, versus "2=(gB3)1=2 and versus �2=�1, for j = 1; : : : ; 10.

In this work, de1ning a mean wave amplitude as

� =[

1T

∫ T

0

(1B2 〈�(x; y; t)2〉

)dt]1=2

(37)

numerical and experimental mean wave amplitude curves versus frequency were com-pared. As a consequence of expansion (10), (37) assumes the following expression:

� =∑i(n;m)

Ii(n;m)

B√T

[∫ T

0(Qi(n;m)(t))2 dt

]1=2

; (38)


where Ii(n;m) is de1ned by

Ii(n;m) ≡√〈cos2 knx cos2 �my〉:

From a numerical point of view, the wave amplitude is calculated applying (38),knowing the time histories of Qi(n;m)(t). Such time histories are known integrating thedynamical system (27) for assigned amplitude and frequency of the forcing oscillation,number and kind of considered modes Qi(n;m)(t). In particular, the number and kind ofconsidered modes — the so-called leading modes — is a crucial point in the integrationof (27). Such leading modes were chosen among the modes which give a signi1cantspectral contribution in the power spectrum of the time histories of the separationsurface. Two values for the amplitude of the forcing oscillation were considered: 2◦

and 3◦.From an experimental point of view, the wave amplitude was calculated applying

(37). In particular, for given values of the frequency and the amplitude of the forcingoscillation, the time history of the separation surface at given points was recordedfor a time T , after the !uid motion became stationary. Then the square of each timehistory was time-averaged and the result was numerically integrated over the surfaceof the container. The numerical and experimental mean wave amplitude–frequencycurves corresponding to the exciting amplitudes 0 = 2◦ and 3◦ are shown in Fig. 4.It is possible to see that the local maxima of the curves appear in correspondence ofthe values of the 1rst three odd resonance frequencies (f10 = 0:23 Hz; f30 = 0:57 Hz;f50 = 0:78 Hz). Even modes are not directly excited by the forcing oscillation butare excited only by nonlinear interactions with odd modes. Experimental observationshighlighted that such interactions can activate even modes either for exciting frequencyvalues in the neighbourhood of the value f10 = 0:23 Hz and for small values of 0

(0 1◦) or for exciting frequency values far from f10 and moderate values of 0

(0 7–8◦):In the following, experimental and numerical time histories of the separation surface

at suitable measurement points and related power spectra, are compared. In particu-lar, the following sloshing con1gurations, characterised by the values of the excitingfrequency and amplitude, were examined:

0 = 3◦; f =

{0:15

0:20Hz ; 0 = 7

◦; f =

0:42 (=f20)

0:57 (=f30)

0:78 (=f50)

0:94 (=f70)

Hz:

Such con1gurations were chosen because they are — but the 1rst two-resonancefrequencies. As it will be shown, the exciting frequency value f=0:20 Hz, very closeto the resonance frequency of the mode (1; 0) (f10=0:23 Hz) generates large amplitudeoscillations (about 0:04 m, in x = 0 m) of the separation surface in correspondence ofsmall values (0 = 3◦) of the exciting amplitude. It is to be noted that the largestamplitude oscillation cannot overcome the oil layer thickness (i.e. 0:06 m) which isthe smallest of the two !uid layers. If the oscillation amplitude of the separationsurface overcomes the value 0:06 m, the separation surface laps the top of the tank


Fig. 4. Plots of the numerical and experimental mean wave amplitude versus frequency, for 0 = 2◦

and0 = 3

◦.

and cannot be represented by the present mathematical model. The exciting frequencyvalue equal to the exact value of the resonant frequency for the mode (1; 0) could notbe adopted, even with an exciting amplitude of 3◦, as the maximum allowed amplitudewould be reached and come over. On the contrary, the subsequent resonant frequenciescould be adopted as exciting frequencies and a moderate exciting amplitude (7◦) was


Fig. 5. (a and b) Numerical and experimental time history of �(x; y; t) and related power spectra.(x = 0:03 m; y = 0:25 m; f = 0:15 Hz; 0 = 3

◦).

needed in order to obtain oscillations of the separation surface of appreciable amplitude.The frequencies f40 = 0:69 Hz; f60 = 0:87 Hz were not considered because, even withmoderate exciting amplitude (7–8◦), the separation surface does not exhibits oscillationsof meaningful amplitude.

In Fig. 5a the comparison between the numerical and experimental time histories ofthe separation surface, in x = 0:03 m; y = 0:25 m is shown. The values of the excitingfrequency and amplitude are, respectively: f=0:15 Hz; 0=3◦. The agreement betweenthe numerical and experimental time histories is satisfactory. In this case the powerspectra of the time histories highlights only a dominant peak in correspondence of


the forcing frequency f = 0:15 Hz; as shown in Fig. 5b; this fact reveals that thesystem is evolving in a linear way, because nonlinear interactions generate peaks incorrespondence of frequencies multiple of the exciting one. The numerical simulationwere obtained considering the modes (1; 0) and (3; 0) which are the 1rst two oddmodes directly excited by the forcing motion. Even modes are not directly excited bythe forcing motion and their excitation is only due to nonlinear interactions which inthis case are negligible.

In Fig. 6a the numerical and experimental time histories of the separation surface areshown for the case f=0:20 Hz; 0 = 3◦; x=0:03 m; y=0:25 m. The amplitude of themotion of the separation surface reaches nearly the maximum allowed value (0:06 m),because the value of the exciting frequency is very close to the 1rst resonance fre-quency. In this case the evolution of the system is nonlinear, as highlighted by thenumerical and experimental spectra (Fig. 6a and b). The experimental spectrum showsfour dominant peaks in correspondence of the frequencies f; 2f; 3f; 4f, which are nearindeed to the resonance frequencies of the modes (1; 0); (2; 0); (3; 0); (5; 0). The numer-ical spectrum obtained considering the 1rst 1ve modes: (1; 0); (2; 0); (3; 0); (4; 0); (5; 0)highlighted a spectral energy distribution not in good agreement with the experimentalone (Fig. 6b). Then a numerical simulation with the 1rst six modes was performed andthe relative spectrum revealed a much better agreement with the experimental spectrum(Fig. 6c). The mode (6; 0) does not give a relevant spectral contribution, but permitsthat the numerical distribution of the spectral energy among the considered modes iscloser to the experimental one.

In Fig. 7a the numerical and experimental time histories of the separation surfaceare shown for the case f=0:42 Hz; 0 = 7◦; x=0:25 m; y=0:25 m: In this work, thisis the only resonant case of an even mode experimentally perceivable. In particular theexciting frequency coincides with the resonance frequency of the mode (2; 0). As theexcitation of the mode (2; 0) by the forcing motion is not direct, but is due to nonlinearinteractions, it is necessary to adopt a forcing motion with moderate amplitude (0=7◦).For the same reason, the numerical simulation had to be performed with at least an oddmode, in order to activate nonlinear interactions. In this case, the maximum amplitudeof the numerical and experimental oscillations is comparable (Fig. 7a). However, dueto the smallness of the oscillations, the noise disturbs the experimental time historiesso that the numerical time history does not reproduce satisfactorily the experimentalone. This case was considered because in this measurement point only even modescontribute to the motion of the separation surface. The mode (1; 0) does not contributedirectly to the motion of the separation surface at the considered measurement point,but only through the energy transfer to the even modes. The presence of even modes isjusti1ed by the analysis of the power spectrum. In fact, in Fig. 7b both the experimentaland the numerical spectrum show three dominant peaks: at f=0:42; 0:84 and 1:26 Hz.Such frequencies are close to the resonance frequencies of the modes (2,0), (6,0) and(12,0). For this reason, the numerical simulation was performed with the modes (1,0),(2,0), (6,0) and (12,0).

In Fig. 8a the numerical and experimental time histories of the separation surfaceare shown for the case f = 0:57 Hz; 0 = 7◦; x = 0:03 m; y = 0:25 m. This case isthe resonance of the mode (3; 0). The numerical simulation was performed considering


Fig. 6. (a–c) Numerical and experimental time history of �(x; y; t) and related power spectra, respectively,considering the 1rst 1ve and six modes. (x = 0:03 m; y = 0:25 m; f = 0:20 Hz; 0 = 3

◦).



◦).

the modes (1,0) and (3,0). The agreement is good both for the time histories andtheir spectra, shown in Fig. 8b. In this 1gures only a dominant peak is revealed, atf = 0:57 Hz; this fact suggests that the regime of motion is almost linear.

In the cases f=0:78 and 0:94 Hz; 0 = 7◦; x=0:03 m; y=0:25 m the modes (5,0)and (7,0) are resonant. In such cases the system exhibits a linear behaviour, becausesingle dominant peaks are revealed by the power spectra (Figs. 9a and 10a). Thenumerical simulations were performed considering respectively the modes (1; 0); (5; 0)and (1; 0); (7; 0) and are in good agreement with the experimental simulations, as it isshown in Figs. 9b and 10b.



◦).

By using a digital video camera, attached to the tank, the examined resonance caseswere 1lmed. In particular, in Fig. 11a–d, instantaneous pictures are shown which cor-respond, respectively, to the following sloshing con1gurations:

0 = 1◦; f = 0:23 Hz (f10); 0 = 7

◦; f =

0:57 Hz (=f30);

0:78 Hz (=f50);

0:94 Hz (=f70):



◦).

In each of these pictures it can be clearly seen the dominant mode corresponding tothe respective resonance frequency. The picture presented in Fig. 11a was obtainedwith an amplitude of 0 = 1◦ in order to have enough time to 1lm the resonancephenomenon. In fact, for the exciting frequency f = f10 = 0:23 Hz after a transientin which the mode (1,0) was clearly visible, the amplitude of the oscillations of theseparation surface became so large that the limit value of 0:06 m were come over. Ofcourse, the duration of such transient depends on the exciting amplitude, in the sensethat the larger is the exciting amplitude, the smaller is the duration of the transient. In



◦).

the other cases examined, the oscillation of the separation surface became stationarywith moderate amplitude and the corresponding mode dominated.

9. Concluding remarks

In this work the motion of two immiscible and superimposed liquids was studied.The con1guration examined consists in a closed, square section tank 1lled with twolayers of immiscible !uids: coloured water and vaseline oil. The tank can rotate around


Fig. 11. (a–d) Instantaneous visualization of the following sloshing con1gurations: (a) 0 = 1◦,

f = 0:23 Hz (f10); (b) 0 = 7◦, f = 0:57 Hz (f30); (c) 0 = 7

◦, f = 0:78 Hz (f50); (d) 0 = 7

◦,

f = 0:94 Hz (f70).

a horizontal axis of an angle =(t). The law of motion (t) is known and characterisedby two parameters: the exciting amplitude and frequency.

Assuming a potential formulation for the absolute !uid motion, suitable expan-sions for the velocity potential functions ’1(x; y; z; t); ’2(x; y; z; t) and separation surface�(x; y; t), the variational formulation, described in La Rocca et al. (2000), was appliedto the present case of sloshing.

The mathematical model was obtained by applying the Lagrange equations to theLagrangian of the motion de1ned as the work done by the pressure over the whole !uiddomain. The unknown, time dependent coe3cients of the expansion of the functions’1(x; y; z; t); ’2(x; y; z; t) and �(x; y; t) can be considered as the generalised coordinatesof the motion. As a consequence of the Lagrange equations, such generalised coordi-nates must satisfy a set of in1nite, nonlinear, 1rst order, ordinary di4erential equations.Considering a 1nite number of generalised coordinates, a nonnormal, singular, dynami-cal system is obtained. The de1nition of a suitable 1rst integral of the motion permittedto put the above mentioned dynamical system in normal form.

In order to take into account for the damping of the motion, dissipative forces— derived from a suitable dissipative function G — were added to the r.h.s. of theLagrange equations. Such dissipative model was validated by a direct comparison withexperimental decay processes. In particular, experimental values for the logarithmic


decrement coe3cients of the 1rst three modes were obtained and compared with thetheoretical values. The agreement among the corresponding values is very good.

The importance of a dissipative model in the numerical simulation is due to thefollowing detected feature of the sloshing: after a transient the spectrum of the exper-imental time history of the separation surface does not exhibit peaks correspondent tothe natural frequencies of the system, unless such natural frequencies are very closedto values multiple of the forcing frequency. In this sense the dissipative model makesthe long time integrations independent from the initial conditions.

The numerical technique used for the integration of the mathematical model consistsof two steps: a predictor and a corrector. During the predictor step, a classical fourthorder Runge–Kutta method is adopted. During the corrector step a technique based onthe Lagrange multipliers permitted to make zero the 1rst integral for each time step.The Taylor series expansion of the coe3cients of the dynamical system was avoidedbecause such coe3cients were numerically calculated for each time step.

Experimental and numerical results were concerned in particular with the time evo-lution of the separation surface at given measurement points. Several resonance caseswere considered, obtaining a good agreement between numerical and experimental re-sults.

Fully nonlinear e4ects were revealed by the case 0 = 3◦; f = 0:2 Hz. In particularthe experimental power spectrum of the time history of the elevation of the separationsurface in x=0:03 m shows the presence of four distinct dominant peaks, correspondingto the multiples of the exciting frequency f and close to the resonance frequenciesof the modes (1; 0); (2; 0); (3; 0); (5; 0). In the other considered cases, the evolution ofthe system is weakly nonlinear in correspondence with the adopted exciting amplitude(0 = 7◦).

Mean amplitude–frequency curves were de1ned by recording the time history ofthe elevation of the separation surface in several measurement points, for 0 = 2◦ and3◦. Such curves were compared with the corresponding numerical ones, revealing agood agreement. In particular, the resonances of the modes (1; 0); (3; 0); (5; 0) werehighlighted by the presence of three local maxima.

In conclusion, the mathematical model, derived by applying the variational formula-tion to the considered problem, constitutes an original mathematical tool in modellingthe sloshing of two immiscible and superposed !uids both in fully and weakly nonlinearconditions.

Appendix A.

A.1. De9nition of n; �m; "nm; !nm

n = n*B; �m = m

*B; "i(nm) =

*B

√n2 + m2

!i(nm) =

√g(�1 − �2)"i(nm) tanh("i(nm)H1) tanh("i(nm)H2)

�2 tanh("i(nm)H1) + �1tanh("i(nm)H2):


A.2. De9nition of �p1(x); �p2(x)

’p1(x)≡ (R + H1)(x − B

2

)+

∞∑k=0

a1k{cosh(41kx) − cosh[41k(x − B)]} sin(41kz) (1)

−∞∑k=0

b1k sin[,1k

(x − B

2

)]sinh(,1kz);

’p2(x)≡ (R + H1)(x − B

2

)+

∞∑k=0

a2k{cosh(42kx) − cosh[42k(x − B)]} sin(42kz) (2)

−∞∑k=0

b2k sin[,2k

(x − B

2

)]sinh(,2kz) (A.1)

where

41k = *1 + 2k2H1

; 42k = *1 + 2k2H2

;

,1k = ,2k = *1 + 2k

B;

a1k =2(−1)k

431kH1 sinh(41kB)

; a2k =2(−1)k

432kH2 sinh(42kB)

;

b1k =4(−1)k

,31kB cosh(,1kH1)

; b2k =4(−1)k

,32kB cosh(,2kH2)

: (A.2)

It is easy to verify that functions (A.1) are harmonic in the !uid domain and satisfythe following properties:

@’p1

@x

∣∣∣∣x=0;B

= R + H1 + z; |z|6 |H1|; (A.3)

@�p1

@z

∣∣∣∣z=−H1

= −(x − B

2

);

@’p2

@x

∣∣∣∣x=0;B

= R + H1 + z; |z|6 |H2|;

@�p2

@z

∣∣∣∣z=H2

= −(x − B

2

); (A.4)

which ensure the ful1llment of the linear boundary conditions (3).


A.3. De9nition of M1i ; M2i ; N1; N2

M1i(n;m) ≡ sinh["nm(H1 + �)]"nm cosh["nmH1]

cos( nx) cos(�my);

N1 ≡ ′′C1(x; �) + ′2[(

x − B2

)�p1(x; �) − G1(x; �)

+12(E1x(x; �) + E1z(x; �))

]+ g

cos()2

�2 − g sin()(x − B

2

)�

+ ′∑n;m

A1i(n;m)(t){(

x − B2

)cos( nx) cos(�my)

×cosh["nm(H1 + �)] − 1cosh["nmH1]

+ nP1nm(�) sin( nx) cos(�my)

− nR1nm(x; �) sin( nx) cos(�my) + S1nm(x; �) cos( nx) cos(�my)} (3)

+12

∑n;m;l;p

A1i(n;m)(t)A1i(l;p)(t){L1nmlp(�)

+ [ n l sin( nx) cos(�my) sin( lx) cos(�py)

+ �m�p cos( nx) sin(�my) cos( lx) sin(�py)]

+M1nmlp(�) cos( nx) cos(�my) cos( lx) cos(�py)}: (A.5)

M2i(n;m) ≡ sinh["nm(�− H2)]"nm cosh["nmH2]

cos( nx) cos(�my);

N2 ≡ ′′C2(x; �) + ′2[(

x − B2

)�p2(x; �) − G2(x; �)

+12(E2x(x; �) + E2z(x; �))

]+ g

cos()2

�2 − g sin()(x − B

2

)�

+ ′∑n;m

A2i(n;m)(t){(

x − B2

)cos( nx) cos(�my)

×cosh["nm(�− H2)] − 1cosh["nmH2]

+ nP2nm(�) sin( nx) cos(�my)

− nR2nm(x; �) sin( nx) cos(�my) + S2nm(x; �) cos( nx) cos(�my)}

+12

∑n;m;l;p

A2i(n;m)(t)A2i(l;p)(t){L2nmlp(�)

+ [ n l sin( nx) cos(�my) sin( lx) cos(�py)

+ �m�p cos( nx) sin(�my) cos( lx) sin(�py)]

+M2nmlp(�) cos( nx) cos(�my) cos( lx) cos(�py)};


where

C1(x; �) =∫ �

−H1

�p1(x; z) dz; C2(x; �) =∫ �

H2

�p2(x; z) dz;

G1(x; �) =∫ �

−H1

(R + H1 + z)@�p1

@xdz; G2(x; �) =

∫ �

H2

(R + H1 + z)@�p2

@xdz;

E1x(x; �) =∫ �

−H1

(@�p1

@x

)2

dz; E2x(x; �) =∫ �

H2

(@�p2

@x

)2

dz;

E1z(x; �) =∫ �

−H1

(@�p1

@z

)2

dz; E2z(x; �) =∫ �

H2

(@�p2

@z

)2

dz;

P1nm(�) =∫ �

−H1

(R + H1 + z)cosh["nm(H1 + z)]

cosh["nmH1]dz;

P2nm(�) =∫ �

H2

(R + H1 + z)cosh["nm(z − H2)]

cosh["nmH2]dz;

R1nm(x; �) =∫ �

−H1

@�p1

@xcosh["nm(H1 + z)]

cosh["nmH1]dz;

R2nm(x; �) =∫ �

H2

@�p2

@xcosh["nm(z − H2)]

cosh["nmH2]dz;

S1nm(x; �) = "nm

∫ �

−H1

@�p1

@zsinh["nm(H1 + z)]

cosh["nmH1]dz;

S2nm(x; �) = "nm

∫ �

H2

@�p2

@zsinh["nm(z − H2)]

cosh["nmH2]dz;

L1nmlp(�) =∫ �

−H1

cosh["nm(H1 + z)]cosh["nmH1]

cosh["lp(H1 + z)]cosh["lpH1]

dz;

L2nmlp(�) =∫ �

H2

cosh["nm(z − H2)]cosh["nmH2]

cosh["lp(z − H2)]cosh["lpH2]

dz;

M1nmlp(�) = "nm"lp

∫ �

−H1

sinh["nm(H1 + z)]cosh["nmH1]

sinh["lp(H1 + z)]cosh["lpH1]

dz;

M2nmlp(�) = "nm"lp

∫ �

H2

sinh["nm(z − H2)]cosh["nmH2]

sinh["lp(z − H2)]cosh["lpH2]

dz: (A.6)


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