Numerical and Experimental Analysis of a FSI Model for a FBR Core: Edge Effects

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Universit`

a degli Studi di Pisa

DIPARTIMENTO DI INGEGNERIA CIVILE E INDUSTRIALE Corso di Laurea Magistrale in Ingegneria Nucleare

Tesi di laurea magistrale

Numerical and Experimental Analysis of a Fluid Structure

Interacton Model for a FBR Core:

Edge Effects

Candidato:

Gianluca Artini

Matricola 436814

Relatori:

Prof. Ing. Walter Ambrosini

Prof. Ing. Nicola Forgione

Dott. Jean-Paul Magnaud

Dott.ssa Lucia Sargentini

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Numerical and Experimental Analysis of a Fluid

Structure Interaction Model for a fbr Core:

Edge Effects

Gianluca Artini

Università di Pisa

Dipartimento di Ingegneria Civile e Industriale

Corso di Laurea in Ingegneria Nucleare

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Abstarct

The work focuses on the fluid flow three-dimensional effects during bi-dimensional oscillation in narrow gaps, with reference to the Phénix core or to a sfr core. For investigating this particular behaviour, two different approaches are used: an experimental approach and a numerical one.

The numerical approach used for evaluating the edge effects is an analysis with the Cast3M upφ code. The upφ model utilizes the Euler linearised equation coupled with the spring-mass dynamic equation. This model is used for representing the experimental facility pise-1a, in order to analyse the three dimensional behaviour and the effects on the oscillating structure.

The experimental approach lies on the exploitation of two exper-imental facilities developed and manufactured by the cea of Saclay: the test-rig pise-1a and the small test facility pise-2c, a hexagonal mono-assembly facility and a two crowns of hexagonal assemblies facil-ity respectively. The aim of the design of these experimental facilities is to reproduce the inertial effects of the Phénix core assemblies. The three dimensional fluid flow (called “jambages”) establishes itself in the inter assemblage space. This fluid behavior is confirmed by some piv visualization on pise-1a. The effects on the structure is a lower frequency vibration with respect to the frequency evaluated by the analytical bi-dimensional models.

Another part of the work concerns the setup of the small scale test facility pise-2c with the calibration and the characterization of its elements for starting the new experimental campaign.

The present thesis work has been carried out during an internship at the cea of Saclay (Paris, France).

Key-Words: hexagonal, sfr, Phénix, fsi, Euler equations, Cast3M, vibrations.

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List of Tables

2.1 _{Design parameters for pise-2c elements . . . 10}

2.2 Support blade geometrical parameters . . . 12

2.3 _{pise-1a element and components weights . . . .} 13

3.1 Experimental frequency obtained for different water heights . . . 26

5.1 Reference parameters for cylindric geometry and free vibration . . . 47

5.2 _{Characteristic data of pise-1a experimental facility . . . 49}

5.3 Frequency evaluated in the numerical simulation by the upφ Cast3M code . . . 49

5.4 Frequency results for three dimensional simulations . . . 53

5.5 Results of the numerical and experimental analysis of free oscilla-tions with different water level in the inter-assembly . . . 55

5.6 Frequency obtained by the simulation with the new definition for the structure stiffness . . . 60

6.1 Calibration curve obtained for element #16 . . . 64

6.2 Sampling parameters . . . 65

6.3 Results of the free oscillation test for 1 mm initial displacement for element # 16 . . . 66

A.1 Calibration Curves for pise-2c elements . . . 70

A.2 Results for the characterization of pise-2c elements for 1 mm initial displacement . . . 70

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Acronym

aurn Arrêts d’Urgence par insertion de Réactivité Négative

fbr Fast Breeder Reactor

fft Fast Fourier Transform

fsi Fluid Structure Interaction

gif Generation Four International Forum

piv Particle Image Velocimetry

pmma Poly-Methyl Meth-Acrylate

sfr Sodium Fast Reactor

Roman Symbols

a Structure Displacement [m] b Gap [m] ca Damping Coefficient c Speed of Sound [m s−1] Ec Kinetic Energy [J] ˆ ei Unitary Vector

fair Frequency in air [Hz]

fwater Frequency in water [Hz]

Ff Fluid Force [N]

g Gravity Acceleration [m s−2]

ks Structure Stiffness [N m−1]

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LIST OF SYMBOLS ix

madd Added Mass [kg]

Madd Added Mass Matrix

ms Structure Mass [kg]

ˆ

n Normal Unitary Vector

p Fluid Pressure [Pa]

r Radial Coordinate u Fluid Velocity [m s−1] xf Fluid Displacement [m]

Greek Symbols

Γ Boundary Domain θ Azimuthal Coordinate

λ Rayleigh Damping Mass Coefficient

µ Rayleigh Damping Stiffness Coefficient

ρ Fluid Density [kg m−3]

σc Neutron Capture Cross Section [b]

σs Scattering Cross Section [b]

φ Displacement Potential Field

ψ Trial Function

ω Natural Pulsation [s−1]

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Chapter 1 Introduction

The nuclear industry research and development sector addresses its attention to the design and development of the new Generation IV reactors. These reactors will be able to obtain the goals, fixed by the Generation iv International Forum (gif) [GIF, 2014], of sustainability and economic features, making the nuclear industry efficient and competitive in the energy production field.

1.1 The GIF and the SFR project

The goals established by the gif for the Generation iv Systems aim to respond to the economic, environmental and social requirements [GIF, 2014]; these goals are eight and they are:

• Sustainability-1: Generation IV nuclear energy systems will provide sustain-able energy generation that meets clean air objectives and provide long-term availability of systems and effective fuel utilization for worldwide energy pro-duction.

• Sustainability-2: Generation IV nuclear energy systems will minimize and manage their nuclear waste and notably reduced the long-term stewardship burden, thereby improving protection for the public health and the environ-ment.

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• Economics-1: Generation IV nuclear energy systems will have a clear life cycle cost advantage over other energy sources.

• Economics-2: Generation IV energy systems will have a level of financial risk comparable to other energy projects.

• Safety and Reliability-1: Generation IV energy systems operations will excel in safety and reliability.

• Safety and Reliability-2: Generation IV energy systems will have a very low likelihood and degree of reactor core damage.

• Safety and Reliability-3: Generation IV energy systems will eliminate the need for off-site emergency response.

• Proliferation Resistance: Generation IV nuclear energy systems will increase the assurance that they are very unattractive and the least desirable route for diversion or theft of weapons-usable materials, and provide increased physical protection against acts of terrorism.

Six nuclear energy systems are identified and selected on the basis of these eight objectives. One of the most promising nuclear energy systems is the Sodium-Cooled Fast Reactor (sfr). The concept of the sfr allows the accomplishment of the goals mentioned above due to the choice of liquid sodium as coolant. In fact, liquid sodium has got some remarkable features like:

• Neutronic proprieties: small capture cross section for fast neutrons σc, high

scattering cross section σs (which allows small leakage) and small energy

loss per collision (negligible moderation) allow the design of a fast spectrum breeder reactor (fbr). A fbr can produce the same, or more, fissile material from the fertile material than the fuel being consumed. Moreover, the fast spectrum choice allows a better radwaste management, through the adoption of a closed fuel cycle, with the fundamental step of the in pile transmutation of long lived fission products. So, the neutronic features of a sfr allow to satisfy the sustainability features of the road map of the new generation nuclear energy systems.

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1.1. THE GIF AND THE SFR PROJECT 3 • Thermophysical proprieties: the most important thermophysical

character-istics of the sodium as a coolant are (at the operating point): – high boiling temperature,

– high thermal capacity, – high density,

– high conductivity, – low vapour pressure,

these characteristics allow the sfr to operate at a pressure near the atmo-spheric one, to have a high heat transfer coefficient, to have low mechanical power requirements for coolant circulation and an high thermal inertia dur-ing operatdur-ing and incidental transients. The passive safety features are in this way enhanced.

• Chemical features: the sodium has got a good compatibility with stainless steel, but it presents a high reactivity in air and water. So, for safety issues, an intermediate loop is needed, between the primary loop with the activated sodium and the water/steam loop.

Several projects for a sfr are under development and the most promising is the French project astrid (Advanced Sodium Technological Reactor for Industrial Demonstration) (fig.1.1b), supported by the Office of Atomic Energy and Alterna-tive Energies (cea), in the framework of the cp-esfr (CollaboraAlterna-tive Project on

European Sodium Fast Reactor) [CP-ESFR, 2014].

However, the sodium cooled fast reactors have got a long history of successfully reactors and prototypes. The most famous are the French pool type reactors Phénix and SuperPhénix and the Japanese loop type reactor Monju.

The Phénix reactor (figure 1.1a) was connected to the grid in the 1974 and definitely shutdown in 2009 with 34 years of experience.

During this period, the Phénix reactor underwent several periods of shutdown. In a period between the 1989 and the 1990, the Phénix reactor was stopped for four emergency shutdowns (scram) due to a negative reactivity [Dumaz et al., 2012].

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(a) The Phénix reactor _{(b) The astrid reactor}

Figure 1.1: The sfr: the Phénix and astrid reactors

At this time, no accidental scenario is completely validated for this type of acci-dents.

These accidents are called aurn (Arrêts d’Urgence par Réactivité Négative), and a typical power time history is shown in figure 1.2.

After some investigations, the only initiating event compatible with the short time scale of the occurrences is the “flowering” of the reactor core: a rapid radial expansions and a following re-compaction of the core assemblies, could be the only possible initiating event of an aurn event.

By the way, causes of the flowering have not been found yet. Several scenarios for justifying the initial event have been investigated, but no one was validated [Dumaz et al., 2012].

First numerical analyses, performed with simplified models to understand this scenario, have not been able to completely validate it. For this reason further investigations, with more accurate tools, are needed.

In this framework, at the “Commissariat à l’énergie atomique et aux énergies alternatives” (cea) in the “Direction de l’énergie nucléaire” (den) at the “Service

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1.2. AIM OF THE WORK 5

Figure 1.2: Two separate registered signals obtained from the neutron chambers during the last two aurn events in Phénix in 1990

de Thermohydraulique et de Mécanique des Fluides” (stfm), a PhD thesis about fluid-structure interaction mechanisms in a sfr core, during sudden release of liquid or gas, is just performed [Sargentini, 2014].

1.2 Aim of the work

The aim of the present work is to analyse the behaviour of the fluid around an hexagonal assembly during oscillations. In particular, investigating the three di-mensional fluid flow effects phenomena is important to understand the fluid effects on the oscillating structure.

For this purpose, a numerical and experimental approach are adopted: piv measurement are made for qualitative and quantitative analysis of the phenomena and numerical analysis is developed using a “upφ” (see later on chapter 4) model for fluid-structure interaction.

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setup consists in the calibration and characterization of the hexagonal elements by free oscillations test in air.

All these analysis and activities will be useful for the future experimental cam-paign with the test facility pise-2c and for the assessment of the upφ model for studying and evaluating the fluid-structure interaction in a sodium cooled fbr.

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Chapter 2 Experimental Facilities

In this chapter, we are going to describe the experimental facilities designed and built at the cea of Saclay for the purpose of analysing fsi mechanisms: the small scale facility pise-2c and the test rig pise-1a [Sargentini, 2014] made up respectively of nineteen hexagonal assemblies, disposed on two crowns, and one hexagonal assembly.

2.1 Aim of the design

The design of these experimental facilities aims to represent the vibration phenom-ena that occur in the Phénix reactor core during an aurn event. The main goal is to describe the momentum transfer between the oscillating elements and the fluid. The facilities will be also useful to validate the numerical models developed for describing the fsi mechanisms.

The starting point is to obtain a representation of a fbr core, like the Phénix’s one; the aim of the design was to obtain elements vibrating in water at the same frequency of the Phénix reactor elements in sodium for a flowering move-ment with a 3.0 mm width gap. The frequency is estimated in about some Hz, [Cardolaccia, 2012]. This will also permits to maintain the aurn time scale.

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2.2 The PISE-2c small test facility

The pise-2c facility (fig. 2.1) is made up of 19 assemblies, composed by an hexag-onal pmma (Poly-Methyl Meth-Acylate) rod and a stainless steel support. The assemblies are disposed in two crowns in an triangular array like in a fbr core. The assemblies of this experimental facility are oriented in order to have a radial expansion as the one occurring during a core flowering event (fig. 2.3). The central element is fixed, but it acts as “injection system”: in each face of this hexagonal element there is a hole at the middle height for injecting fluid (fig. 2.2). Every el-ements are screwed at their base and all the elel-ements are located in a containment made of pmma measuring 1000x1000x1160 mm (fig. 2.1b).

(a) Experimental facility pise-2c: the 19 elements disposed on two crowns

(b) The complete experimental facility pise-2c with its containment

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2.2. THE PISE-2C SMALL TEST FACILITY 9

Figure 2.2: Simplified view of the injection system for pise-2c facility

The design of pise-2c aims to represent a part of the Phénix reactor core. The purpose of the design was preserving the same time scales of the oscillations in the fluid and the fluid reaction on the structure, in term of added mass. Every pmma assembly is designed in order to preserve the same frequency in water (as in the experimental facility) of a Phénix assembly in sodium, with a 3.0 mm width gap. The frequency of a Phénix assembly during a flowering movement is in order of some hertz (2 ÷ 3 Hz) as seen in [Cardolaccia, 2012]. For this reason, full length-scale and geometry in the cross section were kept the same as the Phénix reactor core element. In fact, the side of the hexagonal assembly is 71.4 mm with a gap of 3.0 mm as in the Phénix reactor. These dimensions, especially the gap size, are the most important for keeping the fluid structure interaction mechanisms unchanged from the Phénix case. On the other hand, for simplicity reasons, the height is only 500.0 mm long.

The assemblies are made of pmma for visualizations reasons of the fluid velocity field during the forseen piv tests.

Concerning the choice of the fluid, sodium can not be used for safety reasons, so we use water. The water density is similar to the liquid sodium density (ρN a =

968 kg m−3, ρH2O = 1000 kg m

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water viscosity is an order of magnitude greater than sodium viscosity. A NaI solution is expected to be used to measure the velocity field; however, the NaI solution density is twice the water density and it will lead to have greater values of the fluid force. Also oil will be adopted in order to enhance and investigate the differences between a non-viscous fluid and a viscous one with also different densities.

The main parameters of the small scale facility’s elements are summarized in table 2.1.

b (mm) 3 Gap

l (mm) 71.4 Side of the hexagon

ρ (kg m−3) 1190 _{pmma Density}

As (mm2) 13251.7 Hexagone surface

Rint (mm) 64.94 Equivalent internal ray

Rext (mm) 67.94 Equivalent external ray

ms (kg m−1) 15.76 Hexagon linear mass

Hhex (mm) 500 Hexagon Height

Table 2.1: Design parameters for pise-2c elements

Every pmma hexagonal rod is equipped with stainless steel support blade. Two kinds of support blade are designed:

• Twin-blades support (fig. 2.4a): it allows the assemblies to vibrate in the perpendicular direction with respect to the blades. In this way the twin-blades simulate the behaviour of a reactor core flowering in the horizontal plane.

• Mono-blade support (fig. 2.4b): it allows the simulation of a cantilever beam behaviour.

The geometrical parameters of the support system are obtained by analysis with the Cast3M code [Cast3M, 2014] and the design values are reported in table 2.2.

The frequency of the structure calculated with the Cast3M code [Cast3M, 2014] is 14.18 Hz in air. Thanks to the knowledge of the ratio between the frequency in air and sodium in the Phénix reactor (fair/fN aI = 6.4) we can obtain a frequency

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2.2. THE PISE-2C SMALL TEST FACILITY 11

Figure 2.3: Cross section of the pise-2c facility. The red arrow shows the radial expansion movement

(a) Twin-blades support (b) Mono-blade support

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H (mm) 500

h (mm) 10

s (mm) 3

L (mm) 220

SD (mm) 84

Table 2.2: Support blade geometrical parameters

In order to have a complete characterization of the behaviour of an hexagonal assembly during fsi event, different kinds of perturbations are considered:

• Free oscillations from a non equilibrium initial position

• Steps fluid injections with different velocity of injection, volume of injected liquid and time range

Concerning the measurement system, all the assemblies of the pise-2c facility are equipped with a strain gauge, glued on the support structure. It is also possi-ble to adopt a non contact vibro-laser-meter to have a different and independent measurement system.

Measurements of the fluid velocity field are expected thanks to the piv tech-nique.

2.3 The PISE-1a experimental facility

At the same time, the experimental facility pise-1a was designed and manufac-tured. This facility is composed by only one pmma hexagonal rod and it keeps the same characteristics of a pise-2c assembly. A pmma hexagonal containment is placed around the rod with a 7.0 mm gap. The gap is two times larger than the Phénix one, but it remains sufficiently small to cause important inertial effects on the fluid.

The hexagonal containment has also got an injection system situated at the middle height.

The hexagonal assembly is equipped with a strain gauge glued on one of the blades; measurements with a vibro-laser-meter are also possible.

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2.3. THE PISE-1A EXPERIMENTAL FACILITY 13

Figure 2.5: The pise-1a experimental facility

The geometrical parameters are the same of a pise-2c elements, mentioned in table 2.1, with the exception of the gap width; the weights of the facility are summarized in table 2.3.

mhex (kg) 7.50 Hexagon mass

mblades (kg) 2.36 Twins-blades mass

mscrews 1 (kg) 0.52 Screws mass

mtot (kg) 10.38 Total mass

Table 2.3: pise-1a element and components weights

The pise-1a facility acts as test rig for the assembly of pise-2c: calibration and characterization of pise-2c’s elements are made with this experimental apparatus. The same experimental campaign of pise-2c will be performed and it will be also used for the validation of numerical models. A piv measurements campaign is made for visualizations of the three dimensional fluid flow effects in the gap region around the pmma hexagonal rod.

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Three-dimensional Effects

In this chapter, we are going to illustrate the three dimensional fluid flow that establishes in the top region of the structure. Thanks to the piv technique it is possible to visualize the fluid flow and the resulting imagines are analysed by a matlab software.

3.1 The Down-Stroke Flow

From the previous experimental and numerical campaign a difference between the experimental value of the added mass and the numerical one [Sargentini, 2014, Angelucci, 2013] was found as shown in figure 3.1.

In fact, from the evaluation of the added mass coefficient proposed by Chen [Chen, 1976], the added mass coefficient, that represents the inertial effects of the fluid on the structure, evaluated by experiments, appears to be lower than the analytical one [Sargentini, 2014, Angelucci, 2013].

fair fH2O = s madd ms + 1 = 2.20 ⇒ C_Mexp ≈ 5.5 fair fH2O = s madd ms + 1 = 3.30 ⇒ C_MChen≈ 10.7 (3.1)

This particular behaviour could be due to the presence of a vertical fluid flow and edge effects. In fact, the fluid flow is sensed to be bi-dimensional only in the

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3.1. THE DOWN-STROKE FLOW 15

Figure 3.1: Added mass coefficient. Experimental and numerical results

central zone of the hexagonal rod as found in literature [Sargentini et al., 2014, Monavon, 2013].

In the top of the hexagonal rod, a vertical flow is produced parallel to the side. This flow is called down-stroke flow or “jambages” (see figure 3.2) [Sargentini et al., 2014, Monavon, 2013].

This vertical flow is due to the constant pressure at the free surface of the fluid. The fluid has the possibility to cross over the top of the hexagon. Due to this preferential way of flowing, the inertial effects produced by the fluid on the structure are lower than the ones predicted by the bi-dimensional theoretical models. In fact, the lower confinement, due to the presence of the down-stroke flow, is the cause of a lower added mass and so of a higher frequency in water, with respect to the predicted one by the bi-dimensional model.

The height in which the fluid flow movement is assumed to be three-dimensional is in the order of the external diameter of the structure [Monavon, 2013]. In our case, this length is about 140 mm; it leads to a height on which the fluid flow is supposed to be bi-dimensional, the so called “2d-height”, about 220 mm high:

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Figure 3.2: Theoretical sketch of a down-stroke flow during a bi-dimensional stru-cuture movement

3.2 Visualization of the Down-Stroke Flow: the

PIV Technique

In order to confirm the presence of the three-dimensional fluid flow, some tests in water are made. The facility used for these tests is the pise-1a test-rig experi-mental facility; for the visualization of the fluid flow we use the piv technique.

Thanks to the comparison of two sequential images taken by a high speed camera shooting at a constant frequency of 500 Hz, we are able to follow the displacement of the particles within the fluid. In making this analysis, we are supposing that the particles, when distributed in a homogeneous way, are moving within the fluid at the same time without perturbing it.

In our work, the piv technique allows us to visualize the fluid flow in different positions, in order to obtain a complete characterization of the fluid displacement and velocity field. Consequently, we make visualizations of the:

• inter-assembly region

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3.2. VISUALIZATION OF THE DOWN-STROKE FLOW 17 • top of the hexagonal rod

• face perpendicular to the movement

3.2.1 Visualization of the Parallel Plane to the Face of the

Hexagon and Perpendicular to the Movement

We visualize the plane at midway of the gap, parallel to the face of the hexagon and perpendicular to the movement of the structure as shown in figure 3.3.

Figure 3.3: Schematic sketch of the experimental apparatus during piv measur-ments for the visualization of the parallel plane to the face of the hexagon and perpendicular to the movement

During the test, the displacement of the structure is measured by the strain gauge glued on the support blades. The displacement is shown in figure 3.4 for a initial displacement a0 = 1.0 mm .

The photos taken by the high speed camera during the free oscillation test, are post-treated with a matlab program. Then we are able to obtain the velocity field

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−1 −0.5 0 0.5 1 0 1 2 3 4 5 Displacement [mm] t [s]

Figure 3.4: Displacement time history during piv visualization of the parallel plane to the face of the hexagon and perpendicular to the movement

Figure 3.5: piv photo of the parallel plane to the face of the hexagon and perpen-dicular to the movement

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3.2. VISUALIZATION OF THE DOWN-STROKE FLOW 19 for the velocity components u and w (horizontal and vertical velocity respectively) and also the velocity profile. In figure 3.6 and 3.7 are represented the velocity field and the velocity profile near the top of the hexagon at time t = 0.1 s obtained by the matlab post-processing.

−20 0 20 40 −140 −120 −100 −80 −60 −40 −20 0 Plane width [mm] Plane height [mm]

Figure 3.6: Post-processed image representing the fluid velocity field The pictures clearly show that the fluid moves to the free surface with an higher axial velocity in the middle of the plane, as described by the model shown in figure 3.2.

From figure 3.7a, in proximity of the hexagonal rod’s head, the vertical velocity is comparable to the horizontal one: it is even higher than the horizontal velocity;

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−40 −30 −20 −10 0 10 20 30 40 −100 −50 0 50 100 150 Plane width [mm] Velocity [mm/s] u w

(a) Fluid velocity profile at the top of the pise-1a hexagonal rod at 0.1 s

−40 −30 −20 −10 0 10 20 30 40 −150 −100 −50 0 50 100 Plane width [mm] Velocity [mm/s] u w

(b) Fluid velocity profile at the bottom of the pise-1a hexagonal rod at 0.1 s

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3.2. VISUALIZATION OF THE DOWN-STROKE FLOW 21 but, from figure 3.7b, the axial velocity is smaller than the horizontal one and even negligible. In the central part of the hexagonal rod, the fluid flow around the hexagonal rod can be reasonably considered bi-dimensional.

As the theory explains [Monavon, 2013], it is possible to consider the fluid flow bi-dimensional when the distance between the plane of the motion and the top of the hexagon is in the order of the external diameter of the structure. In our case, as above said, the distance where the three-dimensional effects are negligible is about 140 mm, as clearly shown by the piv post-treated image.

The velocity profile agrees with the theoretical analysis presented in literature see [Sargentini et al., 2014, Sargentini, 2014]: the boundary layer is near the wall.

3.2.2 Visualisation of the Perpendicular Plane to the

Inter-assembly and Parallel to the Displacement

We visualize the plane perpendicular to the inter-assembly and parallel to the displacement at the top and at the bottom (near the twin-blades support) of the pmma hexagonal rod (fig. 3.8).

Figure 3.8: Schematic sketch of the experimental apparatus for piv for the vi-sualization of the perpendicular plane to the inter-assembly and parallel to the displacement

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It is a very important visualization test, because it allows us to verify if the vertical fluid flow is symmetric. Only a qualitative analysis is made. The velocity field visualized is made of the v, w velocity components. In figures 3.9a and 3.9b the velocity field in the nearby of the hexagon head is presented.

In the top part of the hexagonal rod, the vortex formation is easily detectable: the vortex is attached to the corner and it follows the structure in its movement. The vortex develops in the totality of the gap region and it penetrates along the side of the hexagon with a length in the order of the hexagon side length.

Otherwise, approaching the central zone of the hexagonal rod, the axial fluid flow becomes negligible with respect to the horizontal fluid flow. This confirms that the fluid flow is bi-dimensional in the central part of the hexagon.

In the bottom part of the hexagonal rod, the presence of the twin-blades modi-fies the fluid flow: the presence of the support prevents the formation of the vortex. Nevertheless, the fluid flow rests in proximity of this region principally axial as in the hexagonal rod’s head. But, moving from the bottom of the hexagonal rod and reaching the central zone, the fluid flow becomes, again, bi-dimensional as seen in the previous case (fig. 3.10).

3.2.3 Visualization of the Plane at the Top of the Head of

the Hexagonal Rod

In order to understand the fluid’s passage from an inter-assembly region to the other, we visualize the plan at the top of the hexagon (3.11a). The fluid exits from the assembly at the top of the figure 3.11b and it enters in the inter-assemblage space on the other side. The movement is symmetric with respect to the perpendicular plan to the displacement.

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3.2. VISUALIZATION OF THE DOWN-STROKE FLOW 23

(a) Fluid velocity field

(b) Close-up of the fluid velocity field in proximity of the hexagonal rod head

Figure 3.9: piv results for he perpendicular plane to the inter-assembly and parallel to the displacement

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2 4 6 8 10 12 14 0.2 0.5 0.8 Plane height [cm] Plane width [cm]

Figure 3.10: Close-up of the fluid velocity field near the bottom of the hexagonal rod: twin-blades effect

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3.2. VISUALIZATION OF THE DOWN-STROKE FLOW 25

(a) Schematic sketch of the experimen-tal apparatus for piv of the plane at the top of the head of the hexagon

0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Plane depth [mm] Plane width [mm]

(b) Fluid velocity field in the top plane of the hexagon rod head

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3.3 Tests with Different Height of Water in the

Inter-assembly Region in PISE-1a

Experi-mental Facility

In order to understand how the fluid’s inertia acts on the structure’s movement we make some tests with different water height in the inter-assembly region. Three different test are made with:

• the water level is equal to the height of the hexagonal rod

• the water level is equal to the middle height of the hexagonal rod • the water level is equal to a quarter of the hexagonal rod height

The displacements measured by the strain gauge are represented in figure 3.12. The frequency of these three tests are obtained by an analysis in the frequency domain and they are shown in table 3.1.

Test Water height fH2O(Hz)

1 Hhex 5.33

2 0.5Hhex 7.84

3 0.25Hhex 10.69

Table 3.1: Experimental frequency obtained for different water heights

From table 3.1 it is easy to see that the frequency in water tends to increase as the level of water decreases. The value of the frequency tends, obviously, to the value of the frequency in air. This shows that the fluid force acting on the twin-blades support is negligible compared to the fluid force acting on the hexagonal rod walls.

In figure 3.13 we report the signal obtained by the strain gauge post-treated with another method: the modal identification method (see [Juang et al., 1988]). The method bases itself on the Eigensystem Realization Algorithm (era). It uses Makarov parameters to form the Hankel matrices. From these matrices, frequen-cies and damping ratios can be determined by Singular Value Decomposition.

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3.3. DIFFERENT WATER HEIGHT TESTS 27 −1.5 −1 −0.5 0 0.5 1 1.5 0 1 2 3 4 5 Displacement [mm] t [s]

(a) Water level at the hexagonal rod head

−1.5 −1 −0.5 0 0.5 1 1.5 0 1 2 3 4 5 Displacement [mm] t [s]

(b) Water level at middle height of hexago-nal rod head

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 Displacement [mm] t [s]

(c) Water level at a quarter height of the hexagonal rod

Figure 3.12: Displacement vs time for different level of water in the inter-assembly space

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(a) Water level at the hexagonal rod head

(b) Water level at middle height of hexago-nal rod head

(c) Water level at a quarter height of the hexagonal rod

Figure 3.13: Displacement vs time for different level of water in the inter-assembly space: Modal identification method results

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3.3. DIFFERENT WATER HEIGHT TESTS 29 Furthermore, the signals of figure 3.12c and figure 3.13c show two different frequencies. The higher frequency is the structure frequency and the lower one is the frequency of the water oscillating in the inter-assembly region. In fact, during the experience it was possible to see the formation of waves on the free surface of the fluid and the installation of instabilities. These instabilities are due to the vibration of the fluid (Faraday instabilities) or due to the aspiration of air when the the hexagonal rod is separating from the confinement external structure (Rayleigh-Taylor instabilities) or due to the collision between two waves.

The test with the water height equal to the hexagonal rod height is made also with the piv technique. In figure 3.14 the velocity field obtained is shown. No instabilities are created at the interface water-air.

−20 0 20 40 −120 −100 −80 −60 −40 −20 0 Plane width [mm] Plane height [mm]

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3.4 Conclusion

In this chapter we presented the results obtained by the piv technique. This technique allowed us to visualize the fluid flow and to establish that the fluid flow is three-dimensional. It can be considered bi-dimensional only in the central region of the hexagonal rod. We are able to visualize the vertical flow produced in the top and in the bottom of the structure, the so called “jambages”, and we experimentally confirm that the region where the fluid has got a three-dimensional flow is in the order of magnitude of the external diameter of the structure as predicted analytically by [Sargentini et al., 2014, Monavon, 2013]. The difference between the theoretical evaluation of the added mass by Chen and the experimental one may be due to this down-stroke flow and to its effects on the fluid confinement. The experiences in which the water height changes in the inter-assembly region confirms this argument. The fluid inertial effects decrease with the water height and so the water frequency increases until it tends to the value in air.

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Chapter 4 The upφ Model for FSI

In this chapter, an analytical and numerical model for evaluating the fluid struc-ture interaction will be presented. The inertial effects will be analysed and a mathematical and physical interpretation will be discussed.

4.1 Analytical Model: Fritz “The effect of

liq-uids on the dynamic motion of immersed

solids”

The elementary model proposed in this section is the result of the work of Frizt [Fritz, 1972] and it allows us to make a simple illustration of the fsi problem and to describe the inertial effects.

4.1.1 Simple Model for a 1 and 2 DOF System and a

In-compressible and Non Viscous Fluid

The fluid structure interaction problem is historically described by the Euler lin-earised equations for the fluid domain and by the spring-mass dynamic equation for the body domain [Fritz, 1972].

A stagnant incompressible fluid is confined in annular cavity between two cylin-drical walls; those walls are movable (fig. 4.1).

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ˆ n r θ a R1 R2 ρ p

Figure 4.1: Two concentric cylindrical walls: the annular region is filled by fluid

The fluid’s dynamic behaviour could be described by the equation of potential flow and the pressure field must satisfy the Laplace equation in the fluid domain with the appropriate boundary conditions:

                         ∇2_{p =} ∂ 2_p ∂r2 + 1 r ∂p ∂r − 1 r ∂2p ∂θ2 = 0 ∂p ∂r _r=R 1 = −ρ¨a1cos θ ∂p ∂r _r=R 2 = −ρ¨a2cos θ (4.1a) (4.1b) (4.1c)

where p is the fluid pressure field, ρ is the fluid density, ¨a is the acceleration of the

moving walls, r and θ represent the polar coordinate system and the subscript 1 stands for the interior wall and 2 for the exterior one.

The boundary conditions represent the continuity between the normal compo-nent of the acceleration of the fluid and of the structure, at the moving wall, in terms of pressure.

An analytical solution is obtained for the pressure and for the displacement field. The pressure field is given by the equation:

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4.1. ANALYTICAL MODEL: FRITZ (1972) 33 p(r, θ, t) = −ρ         ¨ a2(t)R22 − ä1(t)R21 R2 2 − R21 r [ä2(t) − ä1(t)]R12R22 R2 2 − R21 1 r         cos θ (4.2)

Now that the pressure field is known, we can find the two components (radial and circumferential) of the fluid’s displacements field:

                 xf,r(r, θ, t) = + " a2(t)R22− a1(t)R21 R2 2− R21 −[a2(t) − a1(t)]R 2 1R22 R2 2− R12 1 r2 # cos θ xf,θ(r, θ, t) = − " a2(t)R22− a1(t)R21 R2 2− R12 − [a2(t) − a1(t)]R 2 1R22 R2 2− R21 1 r2 # sin θ (4.3a) (4.3b)

The velocity and acceleration fields for the fluid domain are found by differen-tiating the displacement field:

         ˙ ~ xf(r, θ, t) = u(r, θ, t) = ∂ ~xf ∂t ¨ ~ xf(r, θ, t) = ∂2x~f ∂t2 (4.4a) (4.4b)

It is important to remark that the problem itself remains a linear problem and the superposition principle is still valid. So, in this case we are able to find the solutions for two different problems, defined by the boundary conditions (ä1 = 1, ä2 = 0) or (ä1 = 0, ä2 = 1) respectively.

The knowledge of the pressure field allows us to evaluate the fluid force acting on the walls. We calculate it by integrating pressure on the inside and on the outside circumferences; the fluid forces Ff 1 and Ff 2 are

         Ff 1 = Z 2π 0 −p(r, θ, t)|_r=R 1R1cos θ dθ Ff 2 = Z 2π 0 +p(r, θ, t)|_r=R 2R2cos θ dθ (4.5a) (4.5b)

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Thanks to equation (4.2), the fluid forces can be written as:                  Ff 1= −ρπR21 R2 2+ R21 R2₂− R2 1 ¨ a1+ ρπR21 R2 2+ R21 R2₂− R2 1 + 1 ! ¨ a2 Ff 2= ρπR21 R2₂+ R2₁ R2₂− R2 1 + 1 ! ¨ a1− ρπR21 R2₂+ R₁2 R2₂− R2 1 + 1 ! + ρπR2₂ ! ¨ a2 (4.6a) (4.6b)

These equations can be written in matrix form as:

Ff(t) =   Ff 1(t) Ff 2(t)  = −   madd −(m1+ madd) −(m₁+ m_add) (m₁+ m₂+ m_add)     ¨ a1(t) ¨ a2(t)   (4.7)

where the coefficients of the matrix, the so called added mass matrix, are defined by the formula:                madd = ρπR21 R2 2+ R21 R2 2− R21 m1 = ρπR21 m2 = ρπR22 (4.8a) (4.8b) (4.8c) It is important to notice that the matrices m1 and m2 represent the fluid mass

that would be contained in the region defined by radius R1 and by the radius R2

respectively; we underline therefore that the difference m2 − m1 represents the

mass of fluid effectively contained in the annular region.

The added mass matrix is a positive-definite matrix. It represents the inertial effects of the fluid over the structure, but, it is important to notice that, it does not represent the real mass of fluid interested in the vibrating phenomena, which is located in the annular region. We can write the term representing the added mass from equation (4.8a) in the following form:

madd = ρπR2

α2_{+ 1}

α2_{− 1} (4.9)

where α is the ratio between the outer surface radius and the inner one.

α = R2 R1

= R + b

R (4.10)

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4.1. ANALYTICAL MODEL: FRITZ (1972) 35

4.1.2 Applying The Fritz Model to 1 DOF System

We consider the external wall fixed and only the internal wall is moving, as a rigid body with a mass ms and a stiffness ks. In this case the natural pulsation of the

system is ω = s ks ms (4.11) The dynamic equation of the structure surrounded by the fluid is the well known mass-spring dynamic equation:

ms¨a + ksa = Ff (4.12)

where Ff represents the fluid force acting on the moving wall. From equation (4.7),

the fluid force due to the pressure field, could be written as:

Ff(t) = −madd¨a (4.13)

This leads to a new form of the dynamic equation for the structure, where the meaning of added mass is well shown:

(ms+ madd)¨a + ksa = 0 (4.14)

In fact, the added mass of the fluid is seen by the structure as a virtual mass oscillating at the same time with the structure.

Figure 4.2 represents the ratio between the added mass madd and the fluid

mass located in the solid domain. From figure 4.2, for α → 1 (case of strong confinement) the term of inertial mass is very important and the inertial effects are predominant; instead, when α → ∞ (case of large confinement), the added mass term tends to the value of ρπR2_{. This value represents the mass located in}

the solid regional and it corresponds to the mass of fluid displaced by the structure. From equation (4.14) it is possible to find the new natural pulsation of the coupled fluid-structure system ω∗

ω∗ = s ks ms+ madd < ω = s ks ms (4.15)

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0 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 ma / ρπ R 2 α Added mass Displaced fluid mass

Figure 4.2: Added mass vs α

which is lower than ω due to the presence of the inertial effects of the fluid on the structure.

4.1.3 Energetic Meaning of the Added Mass Term

The kinetic energy of the fluid, during the vibrating movement is added to the kinetic energy of the structure and, precisely, the added mass term is related to the kinetic energy added to the moving structure. In fact, the kinetic energy Ec

can be evaluated by the formula:

Ec(t) = 1 2 Z R2 R1 Z 2π 0 ρku(r, θ, t)k2r drdθ (4.16) where the velocity field components ur and uθ are obtained by differentiating

equation (4.3a) and (4.3b):

ur(r, θ, t) = − ˙a(t) α2_{− 1} 1 − α2_R2 r2 ! cos θ uθ(r, θ, t) = + ˙a(t) α2_{− 1} 1 − α2_R2 r2 ! sin θ (4.17)

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4.2. NUMERICAL MODEL: THE CAST3M CODE 37 The kinetic energy is therefore:

Ec(t) = 1 2ρπR 2α2+ 1 α2_{− 1}˙a 2 (t) = 1 2madd˙a 2 (t) (4.18)

which defines the energetic interpretation of the added mass: the kinetic energy given to the structure by the fluid is not due to the total mass of the fluid, but only by the added mass i.e. only by the mass taking part in the inertial effects [Fritz, 1972, Broc and Sigrist, 2012].

4.2 A Numerical Model for FSI Implemented in

the Cast3M Code

In this section we are going to discuss the upφ numerical model implemented in the Cast3M code [Cast3M, 2014]. The variational form of the problem will be presented with particular attention to the coupling operator at the solid-fluid

boundary. Also the Rayleigh damping model will be presented for describing

damping effects.

4.2.1 Analytical Equation Set for FSI

The model adopted is an upφ model applied to a generic geometry: the structure has got a boundary surface Γs with normal given by ˆn and the displacement of the

structure is given by a = ~a · ˆei (see figure 4.3).

The basic hypothesis for this formulation are: • For the fluid:

– Incompressible fluid

– No viscous effects of the fluid • For the structure:

– Mono-dimensional motion of the structure – Small amplitude oscillations

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Figure 4.3: A pipe wetted by fluid

Also, no damping coefficient ca is taken into account for the structure.

The fluid is governed by the Euler linearised equations and the structure by the mass-spring dynamic equation.

             ∇ · ~u = 0 ρ∂~u ∂t = −∇p ms~a + k¨ s~a = ~Ff (4.19a) (4.19b) (4.19c)

The term ~Ff represents the force exchanged between the fluid and the structure

at the interface, that, under the hypothesis of no viscous effects, is given by only the pressure field:

~ Ff ∝

Z

Γ

−pˆn dΓ (4.20)

For closing the problem, we need to declare the boundary conditions for the fluid. The boundary conditions are the equivalence of the normal component of the fluid velocity and the normal component of the structure velocity. The boundary condition imposes on the fluid in contact with the moving wall to move at the same velocity of the structure. The condition is so a no slip condition for the fluid velocity.

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4.2. NUMERICAL MODEL: THE CAST3M CODE 39 The boundary condition can be also assigned as equivalence of accelerations for the solid structure and for the fluid at the moving wall. In this way, the boundary condition is:

~¨a · ˆn = ~˙u · ˆn (4.22) Thanks to the momentum balance equation (4.19b), we obtain:

∇p = −ρ∂~u ∂t → ∇p · ˆn|Γs = − ρ ∂~u ∂t · ˆn _Γ s = − ρ ~¨as· ˆn _Γ s (4.23)

So the boundary conditions can be also defined in terms of pressure:

~¨a · ˆn = ~˙u · ˆn = 1

ρ∇p · ˆn (4.24)

4.2.2 Cast3M Code Equation Set

The base equation set implemented in the Cast3M code is different from the equa-tion developed for fsi proposed by [Fritz, 1972]. The fluid is not assumed as incompressible, but slightly compressible. This characteristic is due to a numer-ical issue: the code solves the problem using the Lagrangian multipliers method (the penalty method) and so it needs the presence of the compressibility in order to avoid an ill-conditioned problem.

Therefore the equations used are:

               ρ∇ · ~u = −p¨ c2 ρ∂~u ∂t = −∇p ms~a + k¨ s~a = ~Ff (4.25a) (4.25b) (4.25c)

4.2.3 Variational form of the governing equations

From the momentum balance equation (4.25b), by applying the divergence oper-ator, we can obtain the following equation:

∇2_{p =} 1

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The variational formulation of the equation (4.26) is obtained by using the trial function ψ: Z Ω ψ∇2p − ψ 1 c2p¨ dΩ = 0 ∀ψ and ∀Ω (4.27) The previous equation can be written as:

Z Ω ¨ pψ dΩ = c2 Z Ω ψ∇2p dΩ = c2 Z Γ ∇p · ˆnψ dΓ − c2 Z Ω ∇p · ∇ψ dΩ ∀ψ, ∀Ω (4.28)

where ˆn is the normal unitary vector at the boundary Γ.

Thanks to the fluid momentum balance equation (4.25a), we can introduce the fluid acceleration in the surface integral

Z Γ ψ∇p · ˆn dΓ = −ρ Z Γ ∂~u ∂t · ˆn dΓ (4.29)

The surface integral can be decomposed depending on the type of surface that we are considering: the interface with the structure’s moving wall or the fixed walls:

Γ = Γext

[

Γs (4.30)

For a fixed surface the integral becomes:

Z

Γext

ψ∇p · ˆn dΓ = 0 (4.31) instead for the structure’s moving surface, thanks to the continuity condition (4.22), the surface integral becomes:

Z Γs ψ∇p · ˆn dΓ = −ρ Z Γs ψ~¨a · ˆn dΓ (4.32)

For the structure, the dynamic equation governing the solid motion is, (ms~¨a + ks~a)ψ = − Z Γs ψpˆn dΓ ψ ∀ψ (4.33)

and its variational formulation with the trial function ψ is:

(ms~ä + ks~a)ψ · êi = ψ ~Ff · êi = − Z Γs ψpˆn dΓ · êi ∀ψ (4.34)

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4.2. NUMERICAL MODEL: THE CAST3M CODE 41 The final coupled system is the following one:

         (ms~ä + ks~a)ψ · êi = ψ ~Ff · êi = − Z Γs ψpˆn dΓ · êi ∀ψ Z Γs ψρ~ä · ˆn dΓs− Z Ω ∇ψ · ∇p dΩ − 1 c2 Z Ω ψ ¨p dΩ = 0 ∀ψ and ∀Ω (4.35a) (4.35b)

For solving the equation system it is necessary to put the analytical form into a matrix form. Thanks to the finite element formulation we can describe each term of the coupled system with his matrix form. The discretization, with the finite element method, of the previous equations leads to define the coupling operator for the fluid-structure interface

Z Γs ψpˆn dΓ → C (4.36a) Z Γs ψρ~¨a · ˆn dΓ → ρCT (4.36b) The mass and stiffness matrices for the fluid are defined by

Z Ω ¨ pψ c2 dΩ → Mf (4.37a) Z Ω ∇p · ∇ψ dΩ → Kf (4.37b)

Instead, for the strucutre the mass and stiffness matrices are

msa → M¨ s (4.38a)

ksa → Ks (4.38b)

In this way, a coupled matrix system is obtained. The unknowns are the

pressure p for the fluid and the displacement a for the structure:

  Ms 0 ρCT Mf     ¨ a ¨ p  +   Ks −C 0 Kf     a p  =   0 0   (4.39)

However, the above formulation is not symmetric and so it is difficult to solve. But it is possible to obtain an equivalent formulation of the problem with symmet-ric matsymmet-rices. This is possible by adding a new variables for the fluid: the potential

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displacement field φ [Broc, 2011].

The potential displacement field is related to the fluid displacement field by the formula:

~xf = ∇φ (4.40)

Through this definition, the fluid displacement field is supposed to be irrotational at the beginning and so also during his motion. Introducing this new variable in the momentum and mass balance equations, we obtain:

       ρ∇ ¨φ + ∇p = 0 ρ∇2φ +˙ 1 c2p = 0˙ (4.41a) (4.41b) that it leads to ¨ φ = −p ρ (4.42)

The final system, the so called upφ, is a symmetric system and it is:

     Ms 0 ρC 0 0 Mf ρCT _M f −ρKf           ¨ a ¨ p ¨ φ      +      Ks 0 0 0 1/ρMf 0 0 0 0           a p φ      =      0 0 0      (4.43)

In this formulation the right-hand member of equation (4.43) could be a non zero vector, representing the volumetric and the external forces acting on the fluid and on the structure. But, in this dissertation, these effects are not considered, for simplification reasons.

The matrix Madd = ρCKf−1CT is the added mass operator. This operator is

symmetric and positive-definite. It describes the inertial effects between a struc-ture and a stagnant an lightly compressible fluid.

4.2.4 Rayleigh Damping Method

As previously said, the upφ model does not take in to account the structure ing. Moreover, as the viscous effects are negligible in this model, the added damp-ing effects are not considered. For obtaindamp-ing a damped structure movement, the

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4.2. NUMERICAL MODEL: THE CAST3M CODE 43 damping coefficient can be evaluated by the Rayleigh method [Broc and Sigrist, 2010].

The Rayleigh formulation describes the structure damping factor as a linear combination of the stiffness and mass of the structure as shown in equation (4.44)

ca= λms+ µks (4.44)

The structure dynamic equation with the addition of the Rayleigh damping factor becomes:

ms~¨a + ca~˙a + ks~a = ~Ff

ms~¨a + (λms+ µks)~˙a + ks~a = ~Ff

(4.45)

The equation set with the damping factor described by the Rayleigh model,

be-comes: _       ρ∂~u ∂t = −λρ~u − ∇p ˙ p = −ρc2∇ · ~u − µρc2∇ · ~˙u (4.46a) (4.46b) The boundary condition remains the same as in the case of a not damped structure:

~

u · ˆn = ~˙a · ˆn (4.47) With the definition of q as

q = −

Z

ρc2∇ · ~u dt (4.48)

it is possible to rearrange the fluid equations as

   ¨ q + λ ˙q = c2∇2_p q + µ ˙q = p (4.49a) (4.49b) The finite element method begins with the variational formulation using the trial functions ψ defined in the domain Ω

Z Ω ψ (¨q + λ ˙q) dΩ = Z Ω ψc2∇2_{p dΩ} _{∀ψ and ∀Ω} _(4.50)

With the boundary condition at the interface solid-fluid and the momentum bal-ance equation for the damped system we can obtain the equation for the fluid

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system: Z Ω ψ (¨q + λ ˙q) dΩ = Z Γ −ρc2 ~¨a + λ~˙a · ˆnψ dΓ − Z Ω ∇p · ∇ψ dΩ (4.51)

The previous equation can be rewritten in its matrix form − ρCT _(¨_{a + λ ˙a) − M}

f(¨q + λ ˙q) − Kfp = 0 (4.52)

where the matrix operators are the same defined in the case of not damped struc-ture (see section 4.2.3). The new definition of φ is

− ρ( ¨φ + λ ˙φ) = q + µ ˙q (4.53)

It can be noticed that , since p = q + µ ˙q, it can be deduced that

− ρ( ¨φ + λ ˙φ) = q + µ ˙q = p (4.54)

The final system is so:

    Ms 0 −ρCT 0 0 Mf −ρCT _M f ρKf         ¨ a ¨ q ¨ φ     + λ     Ms 0 −ρCT 0 0 Mf −ρCT _M f ρKf         ˙a ˙ q ˙ φ     + + µ      Ks 0 0 0 1 ρc2Mf 0 0 0 0          ˙a ˙ q ˙ φ     +      Ks 0 0 0 1 ρc2Mf 0 0 0 0          a q φ     =     0 0 0     (4.55)

where the damping matrix is constitued by a linear combination of the mass and stiffness martrix, as the Rayleigh damping method proposes.

4.3 Conclusion

The description of the inertial effects needs the computation of the added mass matrix. The evaluation of the added mass matrix needs, in turn, the description and the meshing of the fluid and the solid domains and also, especially for the

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4.3. CONCLUSION 45 fluid/structure operator, of the contact zones between the fluid and the walls.

The complexity of geometries in nuclear applications, like sg or reactor core, makes the modelling and the meshing very difficult and heavy.

The upφ method explained in this chapter is valid for a lot of different config-urations, but one of its most important limitations is to describe only the inertial effects for a stagnant fluid. Another major limitation is that it considers a non vis-cous fluid: this hypothesis is valid only for “small displacements” of the structure, for which case fsi is characterized only by “inertial effects”.

The phenomena dissipative effects can be taken in to account thanks to the Rayleigh damping model [Broc and Sigrist, 2010]. More complete and efficient models are expected from a r&d program on the homogenization of the Navier Stokes equations.

Nevertheless the use of the Rayleigh damping model leads to solve a problem with additional terms to the Euler linear equation. The term µρc2∇ ·~˙u in the mass

equation (4.46b) corresponds to a volume viscosity. The term in the momentum balance equation (4.46a) −λρ~u in the case of a fluid in rigid vessel, for example,

corresponds to a force applied to the fluid by the vessel.

However the basic assumption and at the same time a limitation of this model still remains the “perfect fluid hypothesis”. An extension of the method discussed herein would be possible thanks to the Navier Stokes equations for the fluid.

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Numerical Analysis of the

PISE-1a Facility

In this chapter we are going to describe the results obtained by the numerical simulations made thanks to the upφ Cast3M code [Cast3M, 2014] of the pise-1a experimental facility.

5.1 Description of the problem

We analyse the pise-1a configuration: a hexagonal rod freely oscillating sur-rounded by a non viscous fluid initially in still conditions, stagnant, in a very confined geometry. The rod vibrates freely starting from an initial nonequilibrium position ao. Thanks to the upφ model, we are able to obtain the pressure and the

potential displacement field for the fluid and the displacement for the structure. At first, a simple bi-dimensional geometry problem is studied and a comparison is made for a cylindrical geometry with the analytical model proposed by Fritz [Fritz, 1972].

After that bi-dimensional and three-dimensional simulations for two concen-tric hexagonal geometries are made and it is compared with the experimental results obtained in the previously experimental campaign by [Sargentini, 2014, Angelucci, 2013].

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5.2. BI-DIMENSIONAL ANALYSIS 47

5.2 Bi-dimensional Analysis

5.2.1 Convergence of the Model for a bi-dimensional

cylin-drical geometry

In order to evaluate the convergence of the model, we make a sensitivity analysis concerning the number of azimuthal (along the perimeter of the structure) and radial elements of the mesh (fig. 5.1). These analyses were done for a reference case; we chose cylindrical geometry, whose parameters are described in table 5.1. This allows us to make a comparison with the analytical model proposed by Fritz. The rod frequency in water is evaluated analytically by the formula of equation 4.15 thanks to the knowledge of the added mass (eq. 4.9)

f_HF ritz 2O = 1 2π s ks ms+ madd = 0.573 (5.1)

Structure Parameters Values

ms kg m−1 10

ks N m−2 395

fair Hz 1

Geometry Parameters Values

R mm 50

b mm 25

Fluid Parameters Values

ρ kg m−3 1000

Table 5.1: Reference parameters for cylindric geometry and free vibration In figure 5.1 the cylindrical geometry is represented and the azimuthal and radial meshes are evidenced.

The evolution of the frequency in water as a function of the radial and azimuthal mesh number is shown in figure 5.2.

We obtain the frequency predicted by Fritz [Fritz, 1972] with 10 elements in the gap region and 100 in the azimuthal direction with an error lower than the

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Figure 5.1: Cylindrical geometry mesh and azimuthal and radial meshes Frequency vs Mesh 0.573 0.5735 0.574 0.5745 0.575 0.5755 0.576 1 10 100 1000 freq (Hz)

radial mesh number

f_water upφ Fritz (1972) 0.573 0.5735 0.574 0.5745 0.575 0.5755 0.576 1 10 100 1000 freq (Hz)

azimutal mesh number

f_water upφ Fritz (1972)

Numerical and Experimental Analysis of a FSI Model for a FBR Core: Edge Effects

Universit`

a degli Studi di Pisa

Numerical and Experimental Analysis of a Fluid Structure

Interacton Model for a FBR Core:

Edge Effects

Numerical and Experimental Analysis of a Fluid

Structure Interaction Model for a fbr Core:

Edge Effects

Gianluca Artini

Università di Pisa

Dipartimento di Ingegneria Civile e Industriale

Corso di Laurea in Ingegneria Nucleare

Contents

List of Tables

Acronym

Roman Symbols

Greek Symbols

Chapter 1

Introduction

1.1

The GIF and the SFR project

1.2

Aim of the work

Chapter 2

Experimental Facilities

2.1

Aim of the design

2.2

The PISE-2c small test facility

2.3

The PISE-1a experimental facility

Three-dimensional Effects

3.1

The Down-Stroke Flow

3.2

Visualization of the Down-Stroke Flow: the

PIV Technique

3.2.1

Visualization of the Parallel Plane to the Face of the

Hexagon and Perpendicular to the Movement

3.2.2

Visualisation of the Perpendicular Plane to the

Inter-assembly and Parallel to the Displacement

3.2.3

Visualization of the Plane at the Top of the Head of

the Hexagonal Rod

3.3

Tests with Different Height of Water in the

Inter-assembly Region in PISE-1a

Experi-mental Facility

3.4

Conclusion

Chapter 4

The upφ Model for FSI

4.1

Analytical Model: Fritz “The effect of

liq-uids on the dynamic motion of immersed

solids”

4.1.1

Simple Model for a 1 and 2 DOF System and a

In-compressible and Non Viscous Fluid

4.1.2

Applying The Fritz Model to 1 DOF System

4.1.3

Energetic Meaning of the Added Mass Term

4.2

A Numerical Model for FSI Implemented in

the Cast3M Code

4.2.1

Analytical Equation Set for FSI

4.2.2

Cast3M Code Equation Set

4.2.3

Variational form of the governing equations

4.2.4

Rayleigh Damping Method

4.3

Conclusion

Numerical Analysis of the