Dynamic analysis of a coupled fluid structure problem with fluid sloshing

The present paper is related to the study of a generic linear coupled fluid/structure problem, in which an elastic beam is coupled with an inviscid fluid, with or without sloshing effects. A previous study [ 18) focussed on added mass effects; the present study is devoted to the coupling effects between fluid sloshing modes and structure with fluid added mass modes. The discretization of the coupled linear equations is performed with an axi­ symmetric fluid pressure formulated element, expanded in terms of a FouRIER series [14]. Various linear fluid model are taken into account (compressible, uncompressible, with or without sloshing) with the corresponding coupling matrix operator. The modal analysis is performed with a MATLAB program, using the non-symmetric LANCZOS algorithm [16). The temporal analysis is performed with classical numerical techniques [10), in order to describe the dynamic response of the coupled problem subjected to a simple sine wave shock. The coupling effects are studied in various conditions represented by several non-dimensionnal numbers [12) such as the dynamic FROUDE number and the mass number, based on the geometrical and physical characteristics of the coupled problem. Comparisons are performed on the coupled problem with or without free surface modeling, with a modal and temporal analysis. Coupling effects are exhibited and quantified; the numerical results obtained in the modal analysis here are in good agreement with other previous studies, carried out on different geometry [3, 15]. The temporal analysis gives another point of view on the importance of the coupling effects and their importance at low dynamic FROUDE numbers. The present study gives and will be completed with a non-linear analysis (for both fluid and structure problems) of the coupled problem, using a finite element and


INTRODUCTION
The numerical simulation of coupled fluid structure problems is a subject of great interest, and many studies were carried out on this topic over the past years [ 13], among which the influence of a fluid free surface on the dynamic of the coupled problem (see e.g. in some recent studies [1 , 6,17]). In the present paper, we study the following generic coupled problem, represented by Fig. (1), and focus on the coupling effect occurring between the fluid sloshing modes and the bending structure modes. The problem is characterized by various non-dimensionnal numbers such as the confinement ratio a = .!!:_ , the filling ratio R 2 = .!._ and the length ratio 17 = .!_ ; the dynamic FROUDE number is h R e g;; defined as: F0 =x --. R Psgl 1.

GOVERNING EQUATIONS AND VARIATIONNAL FORMULATION FOR THE DYNAMIC PROBLEM
The structure problem is described in terms of displacement; the governing equations are the following ones (the description includes the dynamic equation, the structure boundary conditions and the coupling condition with the fluid problem):

(I)
The fluid problem is described in terms of pressure [14]; the governing equations are (the description includes the local equation, the fluid boundary conditions and the coupling condition with the structure): =0 8z (z=O) ar (r=R1 02U =-p -case Or (r=R) F 8 t 2 (2) The free surface condition can be described with or without gravity waves, that is: in the former case and: in the latter case.
The variationnal formulation of the dynamic problem is the following one; find u( t ) and p ( t ) such as for all admissible virtual fields of displacement and pressure ( &, t5 p) with the initial conditions for the pressure and displacement field : The various linear and bilinear forms used in the above formulation are the following ones.

FINITE ELEMENT DISCRETIZATION
The discretization of the variationnal formulation is performed with a finite element approach. Since the coupled problem is geometrically axi-symmetric but the dynamic loading is non axi symmetric, the pressure is expanded into a FOURIER serie as 1 : p(r,B,z)= p0(r,z) + �:>.(r,z)cos(n B) + LPm(r,z)sin(mB) (7) n:2: 1 m:2: 1 The 3D fluid problem (r,B,z) is then turned into a 2D problem (r, z) . The two-dimensionnal fluid domain is meshed with four nodes linear finite element. Figure (2) gives a representation of the fluid reference element.
The discretization of the structure problem is performed with two node finite element with two degrees of freedom u, au I fJz and cubic shape functions [5].
The coupled dynamic fluid/structure problem is then written in the non-symmetric form: where R is the coupling matrix, which discretizes r ( . , . ) .
When compressibility effects are taken into account in the fluid problem, he coupled problem is: A MA TLAB program is developed in order to perform the numerical analysis. This program is validated with the comparison between calculated eigenvalues of a fluid problem with free surface and analytical eigenvalues.

MODAL ANALYSIS OF THE COUPLED PROBLEM
The modal analysis of the coupled problem cannot be performed with the mass and stiffuess matrix given in Eq.
After static condensation, the system is reformulated in terms of U and P0 • The corresponding modal problem is then: The coupled problem has the same structure as the initial problem given by Eq. (13): the non-symmetry is typical of a displacement/pressure formulation [ 14,16]. Equation (16)  -/ the fluid sloshing without structure coupling effects the problem to solve is: (17) 2 The validation case is that of a cylindrical tank for which the analytical eingenvalues are given by [8], p. 288. Table ( the structure problem coupled with the fluid without fluid sloshing effects; the problem to solve is: The non-symmetric problem given by Eq. (16) is solved using the LANczos algorithm adapted to non-symmetric problems (16]. Table (2) compares the computed eignevalues with the lanczos function developed in the MATLAB code and the theorical eigenvalues for the non-symmetric example proposed in [16]; this comparison validates the numerical calculation of eigenvalues for generalized non-symmetric problem with our numerical code.    Table 3. Structure mode for FD-100 and FD=10 Tables (4) to (7) show the eigenfrequencies of the fluid problem with and without structure coupling, the structure problem coupled with fluid with or without fluid sloshing, with F0 = 100 and    Table 7. Uncoupled and coupled fluid structure mode with or without fluid sloshing for FD-10, MA=8 and 1=100% For a high dynamic FROUDE number, the fluid and structure coupled modes are practically uncoupled. The fluid eigenfrequencies are slightly increased as well as the structure eigenfrequencies with sloshing compared to the eigenfrequencies without sloshing. These observations are made for all fluid filling ratios. In this case, a separate modal analysis for fluid and structure problem can be performed.
Tables (8) to (11) show the eigenfrequencies of the fluid problem with and without structure coupling, the structure problem coupled with fluid with or without fluid sloshing, with F0 = 10 and    Table 11. Uncoupled and coupled fluid s t ructure mode with or without fluid sloshing for FD-1, MA=8 and 1= 100% For a low dynamic FROUDE number, some fluid and structure modes are likely to be coupled. When an eigenfrequency of the structure coupled with a fluid without sloshing is near an eigenfrequency of fluid sloshing, a coupling effect can be observed. 3 This coupling effect is characterized by the following facts: -/ the sloshing mode eigenfrequency is decreased when coupled with the structure; -/ the structure mode eingenfrequency is raised because of the fluid added mass is lower due to sloshing effect; -/ the sloshing modes above the coupled one are increased whereas the sloshing modes under the coupled one are decreased.
In this case, an uncoupled analysis is not valid. The previous observations are more significant for a low mass number, as shown by Tab. (12).     The influence of confinement on the coupling effect is given by Tab. (13), which compares the ratio e for the fluid and structure eigenfrequencies with or without sloshing (for the structure) and 3 These effects of the coupling between f l uid mode and structure mode in the case of a free surface have been observed by many authors for other fluid-structure coupled problems (see e.g. [3), and [15)).
with. or without structure coupling (for the fluid), for two different values of a . The coupling effect is amplified by confinement (i.e. for small values of a ).   The effects of coupling between fluid modes and structure modes can be represented in the following simple model with a 2D acoustic fluid and a single degree of freedom structure problem, as represented by Fig. (6).
When the fluid is supposed uncompressible, F(m) =I. Equation (19) can be solved with Fig. (7), which plots F and G in the following cases.

DYNAMIC ANALYSIS
A temporal analysis is performed on the structure problem, the fluid problem (i.e. with inner and outer cylinder considered as rigid walls) and the coupled fluid/structure problem, subjected to a sine wave shock. The imposed acceleration is given by: where -r = ...!_ is the shock duration. The discretized equation of ·x r M f.
to. M = --2• " 2 7!f is the first structure The curves are of course the same for the structure mass and stiffness corresponding to the two cases presented in Tab. (l ): for a given problem (geometry and boundary conditions) the reduced displacement and acceleration only depend on the imposed motion [11].

Fluid problem
The dynamic equations of the fluid problem are discretized with finite element technique, as described in §.3, to obtain the dynamic system: The fluid mass terms takes into account the sloshing and acoustic modes of the fluid.
The analytical solution of the uncompressible fluid problem with DIRICHLET conditions for the free surface is given by: ]· ( -l)"cos(q . z).  (1.  The curves exhibit resonance effects in the low (sloshing effects) and high (acoustics effects) reduced frequency range; in the medium frequency range, the fluid forces are governed by added mass effects.

Coupled fluid/structure problem
The discretization of the dynamic equations of the coupled problem with imposed motion leads to the following system, in which V denotes the unknown displacement for the structure problem and P represents the fluid pressure.   In the low reduced frequency range, the calculated displacement is higher with Fv = 10 than Fv = 100 , due to coupling fluid sloshing and structure modes. In the medium and high frequency range, the computed displacement for the two dynamic FROUDE numbers are equivalent: pure added mass effects and acoustic effects have the same influence on the two systems.   A modal and temporal analysis is performed on the coupled problem with or without taking into account the fluid gravity waves, for various dynamic Froude numbers defined with the geometrical and physical characteristics of the coupled problem ..
For low dynamic Froude number, coupling effect between fluid sloshing modes and structure bending modes occur: the first frequencies of the coupled problem calculated with or without dynamic free surface effects can then be very different. The influence of other parameters such as the mass number and the confinement ratio is also exhibited on the coupling process.
A temporal analysis is also performed on the system, when subjected to a simple sine wave shock of given amplitude and duration.
Coupling effects are also illustrated by a comparison of the structure displacement under shock in various conditions. Future study will focus on the non-linear sloshing effects by using a finite element and finite volume numerical coupling procedure.