[Paper Review] An exact pressure evolution equation for the incompressible Navier-Stokes equations

Massimo Tessarotto, Marco Ellero+3 more|ArXiv.org|Dec 7, 2006

Advanced Thermodynamics and Statistical Mechanics3 references3 citations

TL;DR

This paper derives an exact, hyperbolic evolution equation for fluid pressure in incompressible Navier-Stokes flows, bypassing the need to solve the elliptic Poisson equation numerically. The equation, derived from a Lagrangian dynamical system framework, enables time-advancement of pressure along fluid particle trajectories, offering a foundation for efficient Lagrangian CFD simulations without asymptotic approximations.

ABSTRACT

In this paper the issue of the determination of the fluid pressure in incompressible fluids is addressed, with particular reference to the search of algorithms which permit to advance in time the fluid pressure without actually solving numerically the Poisson equation. Based on an inverse kinetic approach recently proposed for the incompressible Navier-Stokes equations we intend to prove that an exact evolution equation can be obtained which advances in time self-consistently the fluid pressure. The new equation is susceptible of numerical implementation in Lagrangian CFD simulation codes.

Motivation & Objective

• To eliminate the need for iterative Poisson solvers in incompressible flow simulations by deriving an exact time-evolution equation for pressure.
• To reformulate the pressure determination problem from an elliptic (Poisson) equation into a hyperbolic evolution equation suitable for Lagrangian methods.
• To provide a mathematically rigorous alternative to asymptotic pressure-based methods such as artificial compressibility or pressure correction schemes.
• To enable numerical implementation in particle-based CFD codes by defining a self-consistent pressure evolution along fluid trajectories.

Proposed method

• Derives a new evolution equation for pressure using an inverse kinetic approach applied to the Navier-Stokes dynamical system.
• Introduces a Lagrangian frame where the fluid pressure evolves along particle trajectories defined by the N-S dynamical system.
• Establishes that the new equation is equivalent simultaneously to the incompressibility condition (∇·V = 0) and the Poisson equation for pressure.
• Uses the convective derivative along particle paths to express the time evolution of pressure without solving the Poisson equation.
• Applies the Euler approximation in time to derive a practical algorithmic form for numerical implementation.
• Validates the approach by showing consistency with the original INSE system under the assumption of strong solutions in C^(2,1) regularity.

Experimental results

Research questions

• RQ1 Can an exact evolution equation for pressure be derived that avoids solving the Poisson equation in incompressible Navier-Stokes flows?
• RQ2 Is it possible to reformulate the pressure determination problem as a hyperbolic evolution equation in a Lagrangian frame?
• RQ3 Can this new equation be numerically implemented in particle-based CFD simulations without asymptotic approximations?
• RQ4 Does the derived equation preserve the incompressibility and momentum conservation constraints exactly?
• RQ5 What are the implications of using effective Mach numbers in the context of the new pressure evolution equation?

Key findings

• An exact evolution equation for pressure is derived that is equivalent to both the incompressibility condition and the Poisson equation, eliminating the need for Poisson solvers.
• The pressure evolution is expressed as a hyperbolic equation along Lagrangian trajectories, enabling time advancement without iterative elliptic solves.
• The equation is valid for strong solutions in C^(2,1) regularity and holds globally in the domain Ω×I under the assumed conditions.
• The method supports the construction of asymptotic solutions via low effective-Mach number expansions, such as the diffusive approximation, for practical numerical implementation.
• The formulation allows for arbitrary initial velocity and pressure scaling, making it suitable for a wide range of flow regimes and numerical schemes.
• The approach provides a rigorous foundation for future Lagrangian particle simulation methods in computational fluid dynamics.