fluids Article Frequency and Amplitude Modulations of a Moving Structure in Unsteady Non-Homogeneous Density Fluid Flow Tolotra Emerry Rajaomazava III 1, *, Mustapha Benaouicha 2,3 , Jacques-André Astolfi 4 and Abdel-Ouahab Boudraa 4   Citation: Rajaomazava, T.E., III; Benaouicha, M.; Astolfi, J.-A.; Boudraa, A.-O. Frequency and Amplitude Modulations of a Moving Structure in Unsteady Non-Homogeneous Density Fluid Flow. Fluids 2021, 6, 130. https:// doi.org/10.3390/fluids6030130 Academic Editor: Iman Borazjani Received: 15 February 2021 Accepted: 18 March 2021 Published: 21 March 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Computed Wing Sail, 151 Boulevard de l’Hopital, 75013 Paris, France 2 Research and Innovation Unit in Naval and Energy Engineering, Segula Tehchnologies, 19 Rue d’Arras, 92000 Nanterre, France; mustapha.benaouicha@segula.fr 3 Cherbourg University Laboratory of Applied Sciences LUSAC-University of Caen Normandy, 60 rue Max-Pol Fouchet, 50130 Cherbourg-Octeville, France 4 Ecole Navale/Arts et Metiers Institute of Technology, IRENav, BCRM Brest, CC 600, 29240 Brest CEDEX 9, France; Jacques-Andre.astolfi@ecole-navale.fr (J.-A.A.); abdel.boudraa@ecole-navale.fr (A.-O.B.) * Correspondence: emerry.rajaomazava@computedwingsail.com; Tel.: +33-6829-894-85 Abstract: A fluid-structure interaction’s effects on the dynamics of a hydrofoil immersed in a fluid flow of non-homogeneous density is presented and analyzed. A linearized model is applied to solve the fluid-structure coupled problem. Fluid density variations along the hydrofoil upper surface, based on the sinusoidal cavity oscillations, are used. It is shown that for the steady cavity case, the value of cavity length L p does not affect the amplitude of the hydrofoil displacements. However, the natural frequency of the structure increases according to L p . In the unsteady cavity case, the variations of the added mass and added damping (induced by the fluid density rate of change) generate frequency and amplitude modulations in the hydrofoil dynamics. To analyse this phenomena, the empirical mode decomposition, a well established data-driven method to handle such modulations, is used. Keywords: fluid-structure interaction; added mass; added damping; frequency modulation; ampli- tude modulation; non-homogeneous fluid density; cavity oscillations; empirical mode decomposition; intrinsic mode functions 1. Introduction Fluid structure interaction (FSI) problems occur when the fluid loading greatly affects the structure’s dynamics and the structure displacement locally affects the fluid flow. Initially studied with simplified models, the simulation of complex coupled problems has developed considerably in recent years. The state-of-the-art in this field is now very mature and several papers with different fields of application domains can be found in the literature [1–3]. The new challenge of FSI problem analysis consists in taking into account complex phenomena, observed both in fluid and solid mechanics, especially in the field of fluids where the dynamics are subjected to many physical quantities such as velocity, pressure, density or temperature. This work focuses on FSI effects in a non-homogeneous density flow. Recent work has pointed out that two-phase flow has an impact on the fluid structure interaction for various devices, such as propeller blades or hydrofoils [4–9]. However, very few published works address the problem of estimating this impact on the structure dynamics. This work is strongly motivated by recent advances in experimental and modeling studies carried out by the authors. It is shown that modal response of the structure could be modified in the presence of cavitation [10]. This modification can be attributed to the presence in the flow of a non stationary liquid-vapor mixture with a strong variation in density at the fluid structure interface. Previous works proposed the decomposition of the fluid variables into two components: the first component is related Fluids 2021, 6, 130. https://doi.org/10.3390/fluids6030130 https://www.mdpi.com/journal/fluids