Exact pressure evolution equation for incompressible fluids

TL;DR

This paper derives an exact pressure evolution equation for incompressible fluids using inverse kinetic theory, enabling pressure computation without solving the traditional Poisson equation, thus advancing computational fluid dynamics methods.

Contribution

It introduces a novel exact pressure evolution equation for incompressible fluids based on inverse kinetic theory, replacing the Poisson equation in fluid simulations.

Findings

Derived an exact pressure evolution equation for incompressible fluids.

Established a dynamical system approach for fluid pressure evolution.

Facilitated pressure computation without solving the Poisson equation.

Abstract

An important aspect of computational fluid dynamics is related to the determination of the fluid pressure in isothermal incompressible fluids. In particular this concerns the construction of an exact evolution equation for the fluid pressure which replaces the Poisson equation and yields an algorithm which is a Poisson solver, i.e., it permits to time-advance exactly the same fluid pressure \textit{without solving the Poisson equation}% . In fact, the incompressible Navier-Stokes equations represent a mixture of hyperbolic and elliptic pde's, which are extremely hard to study both analytically and numerically. In this paper we intend to show that an exact solution to this problem can be achieved adopting the approach based on inverse kinetic theory (IKT) recently developed for incompressible fluids by Ellero and Tessarotto (2004-2007). In particular we intend to prove that the evolution of the fluid fields can be achieved by means of a suitable dynamical system, to be identified with the so-called Navier-Stokes (N-S) dynamical system. As a consequence it is found that the fluid pressure obeys a well-defined evolution equation. The result appears relevant for the construction of Lagrangian approaches to fluid dynamics.