## Mathematical methods and modeling in fluid mechanics

Mathematical methods for fluid mechanics cover a broad range of needs and typically require to use, adapt, or develop advanced techniques of solution. Theoretical bases are required to face the solution of special problems (vortices, boundary layers, invariance properties of parametric models, fluid-solid interaction, to cite a few). Numerical methods in fluid mechanics present a continuous need of evolution to face the increasingly complexity of problems to be solved. Therefore techniques for properly handling irregular geometries, and moving elements are needed, as well as the development of multi-scale methods. This is particularly true when industrial processes are investigated. Finally, mathematical methods and models are the unavoidable tool to analyze large experimental and numerical datasets, and to produce the synthesis process required for understanding physical processes.

Proposed research projects include the following topics.

**Turbulent, thermal boundary-and transition-layers above and within porous media**

Many environmental and industrial situations involve thermal, turbulent boundary- and transition - layers above and within porous bodies. These include gravel beds of natural streams, atmospheric boundary layers over snow or vegetation, bio-heat transfer in human tissues (Khaled & Vafai, 2003), the development of innovative and compact heat exchangers (Kasaeian, Daneshazarian, Mahian, Kolsi, & Chamkha, 2017) and many others. When a turbulent flow takes place through and above a slab of porous, solid material, and a temperature difference is maintained between them, a thermal, turbulent boundary layer develops at the interface between the solid and the free-fluid regions. The surface of the porous material splits the whole flow domain into a surface and a subsurface flow region. When either of these two flow regions exists on its own, flow characteristics are reasonably well understood. Indeed there is a large body of literature on both boundary layers over rough (and smooth) impermeable walls and traditional porous media flows. This is not the case whenever surface flows and porous media interact. A so called transition layer develops in the upper part of the subsurface flow where mean velocities deviate from the typical constant profile of undisturbed porous media. Moreover, within the surface flow, some authors have noticed that the wall permeability can significantly affect the flow resistance of both laminar and turbulent boundary layers (Kuwata & Suga, 2017). The interplay between the thermal properties of the fluid and the porous layer affect the intensity of the turbulent, thermal fluctuations at the interface and, consequently, impact on the resulting interfacial heat transfer rate. The actual values of the equivalent thermal conductivity and the equivalent, specific heat capacity of the porous matrix determine whether there is a substantial thermal equilibrium between the matrix and the interstitial fluid. The proposed project aims to characterize the turbulent, thermal boundary layer developing at the interface between a free fluid, namely a gas species with Prandtl number close to 1, and a porous matrix and to contribute to the advancement of closure models for the volume-averaged energy equation. The assumption of thermal equilibrium between the fluid and the solid matrix will be critically addressed and the error induced by this approximation will be related to the thermal and hydraulic properties of the porous layer. Focusing on relatively low Reynolds numbers, we propose to carry out micro-scale, direct numerical simulations (DNS henceforth) of turbulent flow and heat transfer on a plane channel partially filled with a synthetic porous matrix consisting of either a regular or an irregular arrangement of regularly- or irregularly-shaped obstructions (e.g., the random geometry YADE model created by (Dyck, 2014)): the details of the flow throughout the inter-granular spaces will be captured (see, e.g., (Piller, et al., 2009), (Piller, Casagrande, Schena, & Santini, 2014)). The results of the aforementioned DNSs will be up-scaled and used as reference to validate and possibly improve existing models for, e.g., the shear stress and the heat transfer coefficient over porous surfaces and to assess the validity of the thermal equilibrium assumption under different combinations of fluid and solid materials.

**Hydroacoustics of cavitation in liquids**

This theme has a large number of applications, from naval propellers to pump impellers and lends itself to the development of research activities. In particular, with regard to the acoustic characterization of bubble implosion, parametrization is required in numerical solvers, since it necessarily constitutes a term of sub-grid. In the current state of the research, it is known that cavitation produces unsteadiness of the gaseous phase with consequent pressure pulsations and this effect can be reproduced relatively easily. What is yet to be determined and characterized is the pressure peak associated with the explosion of the gas bubbles, and the effect of this peak on the radiated noise. The project includes a part of the study of the problem, a part of the development of a theoretical model and application of this model to archetypal cases of literature and, subsequently, the development of a numerical algorithm for the reproduction of acoustic pressure peaks in CFD models of type Large Eddy Simulation. Finally, the complete model will be applied to the study of problems of relevance in the literature. Implementation will be carried out within the OpenFoam platform.

**Hydroacoustics in presence of variable density flows**

Literature mathematical models have been developed for modelization of the far field noise radiated by marine media (in particular naval propellers or maneuver surfaces of submarines). It is well known from the scientific literature that when acoustic waves propagate in non-homogeneous media (for example in water columns characterized by vertical variations of temperature and, therefore, of density) they undergo a distortion and a partial reflection in correspondence with regions of density variation. The mathematical models of the hybrid type present in the literature (typically the Ffowcs Williams and Hawkings equation) were developed in the hypothesis of propagation in a homogeneous medium, and, in literature, there is no extension of this equation, or an alternative formulation , able to analyze the propagation of acoustic waves in non-homogeneous media.

The aim of the research project is the formulation of an equation for the propagation of acoustic waves in the presence of flows with variable density, the numerical implementation in CFD solvers of the Large-Eddy Simulation type and the study of physical processes of interest. The ideal candidate has a background in applied mathematics, physics or engineering.