Advection-Diffusion of Scalars with the Incompressible Schrödinger Flow

The description of fluid motion is typically carried out either considering continuum matter or collision models (Lattice Boltzmann methods), with the former being the most common: in this framework, differential balance equations are derived from conservation principles and are solved to obtain density, velocity, pressure and temperature fields. If viscous forces are neglected, the fluid behaves as inviscid, and it obeys the Euler equations: their direct numerical simulation is challenging because of their mathematical structure, which imposes tight constraints on the mesh size and the temporal discretization. Thus, powerful computers are needed to deal with the long computational times of direct numerical simulations of fluids, and still, computational times are often unsuitable for multi-query scenarios and parameter sensitivity studies. A third approach to dealing with inviscid fluids was born in the 1920s when Madelung proposed its interpretation of quantum mechanics based on an analogy between the Schrödinger equation and the compressible Euler equations obtained by adopting a suitable coordinate transformation. This framework has been recently analysed for simulating incompressible inviscid flows, providing a novel methodology for solving the Euler equations, known as Incompressible Schrödinger Flow. Its different mathematical formulation allows a less tight numerical discretization; investigating this approach can give a new perspective on the description of fluids. This work focused on the application of this method in the advection-diffusion of scalars (e.g., temperature or concentration) in inviscid incompressible flows, adopting as velocity that computed by the Incompressible Schrödinger Flow; furthermore, it will be investigating the possibility of inserting into the “quantum” counterpart equation a potential term related to gravity and buoyancy forces.