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\section{Definition of sloshing}
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        "\textbf{Sloshing} means any motion of the free surface inside a container."\cite{Liquid_sloshing_dynamics:_theory_and_applications}
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In a more simple way, sloshing can be pictured in everyday life because it occurs each time one walks with a glass partially or completely filled with water: the liquid can spill out of the container if its free surface is excited enough. Depending on a series on parameters, such as the type of disturbance, the container shape, the frequency of external excitation or the presence of dumping element in the container itself, the free surface can move differently by kind of motion: simple planar or non-planar waves, rotational waves, symmetric or asymmetric modes, periodic or chaotic motion. Things starts to get more complicated when the free liquid surface interacts with elastic container's walls or internal frames, showing up a wave-structure interaction in the form of interacting modal modes.\\
The sloshing processes introduce larger loads on the walls due to the free-liquid surface motion. Containers' integrity due to sloshing phenomena needs to be properly investigated and the typical sloshing parameters need to be known. In this scenario, hydrodynamic pressure distribution, natural frequencies of the free liquid surface and mechanical estimation of stress distribution on structures need to be estimated. The hydrodynamic pressure has two components: the first proportional to the acceleration of the tank; the second one due to the free-surface-liquid motion. Such as any other dynamic problem, mass-spring-damper systems or pendulum are used to extract this information. As written in Ibrahim's work \cite{Liquid_sloshing_dynamics:_theory_and_applications}, despite an infinite number of natural frequencies of the system, only the first ones are really interested in the majority of the engineer applications.\\
Sloshing is a problem particularly sensitive to the initial conditions and boundary conditions, the surface tension, the height of free-liquid surface in the containers (partially or completely filled). The high sensibility leads to hard predictable non-linear effects, and the very problem is that each type of moving containers has its own non-linearity associated.
\section{History of sloshing}
In order to write a quick but essential overview on the impact of sloshing studies in research history, \cite{Liquid_sloshing_dynamics:_theory_and_applications} was used as starting point, because it's one of the most comprehensive work that collects biggest achievements in the literature so far (from 1960 to 2005). From the author Ibrahim it's well known also the review article in \cite{Recent_Advances_in_Liquid_Sloshing_Dynamics}, whose purpose is to review the research work developed in different engineering applications.\\
The issue of liquid sloshing in both moving and stationary containers continues to be a significant focus for aerospace, civil, and nuclear engineers, as well as physicists, road tanker and ship tanker designers, and mathematicians. Civil engineers and seismologists have investigated the effects of liquid sloshing on large dams, oil tanks, and elevated water towers during ground motion. They have also utilized liquid tanks on the rooftops of buildings to mitigate building oscillations caused by earthquakes.\\
Sloshing is a matter of fundamental interest when earthquakes and tsunamis effect on containers is studied. The Great Alaskan earthquake in 1964 can be considered a turning point in recent engineer history: \cite{great_Alaska_earthquake_and_tsunamis} highlights how that tragedy can be considered as one of the strongest stimuli that led to major breakthroughs in earth science research. Sloshing become a matter of interest when associated to seismic events since then, as cited in \cite{Water_Storage_Tank_Response_to_Earthquake_Loads_in_Alaska} about the water storage tank response to earthquake loads. Events that involved the Fukushima I Nuclear Power Plant during the magnitude 9.0 Tohoku earthquake and tsunami in March 2011 represent a pivotal point in the current matter.\\
Since the early 1960s, considerable concern about the dynamics of free-liquid surface motion was expressed also in terms of the impact of liquid propellant sloshing on the flight performance of jet vehicles, leading to the emergence of new areas of research. Abramson (2003) is one of the most influential author in that particular research sector, and reviewed its own work in \cite{Dynamics_of_contained_liquids:_a_personal_odyssey}. It's usual to refers to \cite{Some_Studies_of_Liquid_Rotation_and_Vortexing_in_Rocket_Propellant_Tanks}, whose conclusion lead to the necessity of providing baffles and dumping devices to allow the maintenance of adequate flow rates. In \cite{Simulation_of_fuel_sloshing_characteristics_in_missile_tanks_by_use_of_small_models} Abramson (1960) compares experimental results of test on smaller models of missile tanks with full-scale ones by the means of a similitude theory, while in \cite{Representation_of_fuel_sloshing_in_cylindrical_tanks_by_an_equivalent_mechanical_model} Abramson (1961) provides a mechanical model for representation of linear fuel sloshing with damping in cylindrical tanks. It's important to notice that even Apollo 11 experienced sloshing during the lunar landing phases. In particular, in the mission transcripts the sloshing was considered the phenomenon responsible for the spacecraft uncontrolled descend, the Landing Point Designator uselessness and the premature latch of the propellant low-sensor \cite{Sloshing_dynamics_investigation_by_means_of_non-intrusive_measurement_techniques}. As reported in the beginning of \cite{Sloshing_of_liquids_in_partially_filled_tanks-a_review_of_experimental_investigations}, the Space Exploration Technologies (SpaceX) launched X Demo Flight 2 by means of Falcon 1 (March 20, 2007) occurs in failure. Propellant slosh is considered the primary contributor to an instability in pitch, yaw and in particular in roll which led to a fatal roll torque. The final consequence was overeating of the engine and flame-out of it because that roll torque was too difficult to stabilized even for the roll control thrusters.\\
Faltinsen can be considered one of the most author in sloshing as well as Ibrahim. \cite{Faltinsen_Sloshing} (2009) presents a complete overview on the sloshing problem, but with a special eye on marine and land-based applications. This is the name one must know when are treated topics such as ocean waves and sloshing related with ship motion. Faltinsen interest in coupling of sloshing and ship motion is presented also in \cite{Coupling_of_sloshing_and_ship_motions} and \cite{Proceedings_of_the_International_Workshop_on_Water_Waves_and_Floating_Bodies}. It's fundamental to make mention of \cite{A_nonlinear_theory_of_sloshing_in_rectangular_tanks}, one of the main reference for nonlinear sloshing theory in rectangular tanks.\\
The last field of interest of sloshing that is reported in the current literature review is about public transportation of liquid tank, such as oil tank, by means of vehicle or railways (in example are referenced \cite{Recent_Advances_in_Liquid_Sloshing_Dynamics} and \cite{Faltinsen_Sloshing}). Here is also cited \cite{Sloshing_effects_and_running_safety_in_railway_freight_vehicles} and \cite{Impact_of_liquid_sloshing_on_the_behaviour_of_vehicles_carrying_liquid_cargo}, which define the most critical conditions for tanks safety by the means of simulation about typical accidents, such as derailment or vehicles accidents.

\section{Sloshing theory}
\input{chapters/theory}\label{parag:sloshing_theory}

\section{Sloshing in nuclear industry}\label{parag:sloshing_in_nuclear_industry}
Earthquake induced sloshing may lead to dangerous effects in the nuclear power industry. Housner (1957), which is one of the most influential author on the dynamic behaviour of accelerated fluid containers \cite{Dynamic_pressures_on_accelerated_fluid_containers}, treats about problems of nuclear power plants in \cite{Earthquake_research_problems_of_nuclear_power_plants} (1966). For the records, the accident of SL-1 reactor in the USA (1961) can be also mentioned in this quick literature review: \cite{Additional_Analysis_of_the_SL-1_Excursion} collocates the extreme impact of coolant liquid on the ceiling of the primary pool within the one of the problem that occurs in that tragedy. Another technical report about nuclear reactors and earthquakes is \cite{Nuclear_reactors_and_earthquakes}, cited by many as a reference to recent history of accidents.\\
Nowadays, the sloshing problem in the nuclear reactors is still a very current topic. Seismic safety has been brought to a primary role after the event that involved the Fukushima I Nuclear Power Plant during the magnitude 9.0 Tohoku earthquake and tsunami in March 2011. With the advent in the close future of the new class of \textit{Generation IV Lead-cooled Fast Neutron Reactor demonstrator}, new studies have been conducted to guarantee safety. The most cited authors in this category of sloshing application are Ibrahim (2005), Faltinsen (2003) and Housner (1966). \cite{Seismic_sloshing_effects_in_lead-cooled_fast_reactors} can be considered a good starting point in the study of sloshing in the this new type of Lead-Cooled fast reactors. From the same author, Jeltsov (2015), also \cite{Parametric_study_of_sloshing_effects_in_the_primary_system_of_an_isolated_lead-cooled_fast_reactor} can be cited as well, because it presents a parametric study of sloshing effects in the primary system of this reactors. According to the sloshing theory, in his works Jeltsov affirms that generally the mitigation strategies for earthquake is to take advantage from seismic isolation system. One of the most dangerous collateral effects is the decrease of the fundamental frequencies in the response spectra of some interesting point of the structure. That shifting brings the system closer to liquid sloshing frequencies.\\
Among other relevant studies, \cite{CFD_and_experimental_investigation_of_sloshing_parameters_for_the_safety_assessment_of_HLM_reactors}, published for the American Nuclear Society in 2017, is cited and treats about a CFD and experimental investigation of sloshing parameters in MYRRHA (\textit{Multi-purpose hYbrid Research Reactor for High-tech Applications}) a lead-bismuth cooled fast reactor, managed by SCK CEN the Belgian Centre for Nuclear Research. It's part with the \textit{Advanced Lead Fast Reactor European Demonstrator} in the \textit{Proof of Augmented Safety Conditions in Advanced Liquid-Metal-Cooled Systems} (PASCAL) framework. More details about the project are in the next Paragraph \ref{parag:PASCAL_project}. 

\section{PASCAL project}\label{parag:PASCAL_project}
\textit{Advanced Lead Fast Reactor European Demonstrator} (ALFRED)\cite{Alfred}, is a project by the European Sustainable Nuclear Industrial Initiative, aimed at developing \textit{Generation IV Lead-cooled Fast Neutron Reactor demonstrator} \cite{Alfred_ansaldo_nucleare} as part of the EU's Strategic Energy Technology plan. Lead by FALCON association (Fostering ALfred CONstriction), formed by \textit{Ansaldo Nucleare}, \textit{Ente per le Nuove tecnologie, l'Energia e l'Ambiente (ENEA)} and the Institute for Nuclear Research \textit{(ICN)}, this project is committed to the future of fission nuclear power as the carbon-free source that, together with renewable, will drive the transition towards sustainable, reliable and affordable electricity. This type of reactor is specific because represents a solution for fast reactors cooled with heavy liquid.\\ 
Von Karman Institute (VKI) has been conducting both experimental and numerical studies in collaboration with \textit{Ansaldo Nucleare} to ensure nuclear safety for operations related to ALFRED.\\
European project \textit{Proof of Augmented Safety Conditions in Advanced Liquid-Metal-Cooled Systems} (PASCAL) \cite{Pascal_project} is well suited for the previous objective. PASCAL project aims to the advancement of the safety research on innovative nuclear reactors ALFRED and MYRRHA, and to generate ready-for-use evidence for the discussions between the designers of those reactors and the respective safety authorities in the pre-licensing phase. It's part of the EU Framework Program for Research and Innovation \textit{Horizon 2020}.\\
Earthquakes induced sloshing may lead to dangerous effects in a nuclear reactor pool during seismic events. PASCAL's framework dedicate to the sloshing part aims to identify if seismic activity is critical for the nuclear reactor safety and to give quantitatively a prediction of the damages on the involved structures. One can think about it as a progression of the previous works that started in the 1960s (Paragraph \ref{parag:sloshing_in_nuclear_industry}).
\section{Thesis project's objective}
The objective of this thesis is to investigate sloshing processes in the ALFRED's model. Model scale experiments (1:18.4) have been performed by means of the von Karman Institute's scaled reproduction in plexiglass of the very hot plenum of the actual demonstrator. The present chapter has been used to introduce an historical background on the sloshing induced accidents and features, some brief hints of linear and weakly linear theory. PASCAL project has been also mentioned because some of the seismic simulation test series have been post-processed in the current thesis.\\
Chapter 2 collects the fundamental characteristic of the experimental setup and the methodology of investigation. Similarities studies, based on equivalence related to Froude number, had already been done in the past years and are quickly reported in this chapter to explain the similitude within distilled water in the reduced model and the molten lead in the demonstrator. The geometry model is then presented to give a glimpse of its own complexity, which affects consistently the dominating sloshing processes taking place. Tests have been set up through usage of the VKI’s three degree of freedom shake table with hydraulic piston type actuators. Capacitance probes have been used as an intrusive method of investigation to measure the amplitude of distilled water waves of the free surface when an excitation is applied. Their calibration is performed using the Monte Carlo method, according to the ASME PTC 19.1 (2018). A complete test matrix of the performed tests is presented.\\
Chapter 3 presents the result of some of the most significant test series which has been performed. Harmonic base excitation in one Degree of Freedom (DoF) has been identified as the starting point of the sloshing processes' investigation. Different water depths, forcing amplitudes and frequencies outside and at/near to the fundamental resonance of the system have been selected to highlight the behaviour of the liquid free-surface when they change. Nonlinear free-surface physics are expected to be identified, combined with a non-negligible wave-structure interaction that adds complexity to the sloshing processes.\\
Chapter 4 presents the results of some test series performed in the sloshing framework of PASCAL, where some idealized norm-base earthquakes input signals have been undertaken. These tests contain base excitation signals outside and close to the fundamental system frequency. The short duration input signals add further to the randomness of the sloshing processes. A hazard condition in the primary nuclear reactor has been simulated by removing three of the internal cylinders (steam generators).\\
As stated in the abstract and the introduction, the framework of the current Master thesis is the experimental investigation of the highly nonlinear sloshing processes that occur in the scaled reproduction of the ALFRED's model. At the end of each of the performed test series, some observations can be done on the nuclear safety due to sloshing induced accidents. Harmonic based excitations have been extremely useful to decouple the complexity of current study case. As a consequence, seismic simulations can be critically analysed.