A model for second-gradient compressible Navier–Stokes fluids: Overview, connections with quasi-incompressible RANS equations, and applications

This paper concerns the second-gradient compressible fluids model and examines the mechanical constitutive equations that yield the Navier–Stokes equations with additional terms accounting for hyperviscosity effects via higher-order spatial derivatives of the velocity field. Some general aspects are reviewed, including the field equations governing momentum and kinetic energy, with particular attention paid to the definition and interpretation of pressure. The correspondence between the resulting field equations and the quasi-incompressible Reynolds-Averaged Navier–Stokes (RANS) equations is established. As a specific application with engineering relevance, the second-gradient model is employed to describe anomalously large viscous dissipation in the propagation of finite-amplitude pressure waves in water-filled pipes, a phenomenon commonly referred to as water hammer waves. Within a one-dimensional (1D) formulation, a wave equation is derived, highlighting the influence of a new dimensionless number that dimensional analysis identifies as crucial for wave propagation and kinetic energy dissipation. Phenomenological parameters are estimated by fitting the analytical solution of the wave equation (derived under appropriate initial and boundary conditions) to experimental data available in the literature. The accuracy of the 1D model is assessed, and its strengths, limitations, and critical aspects are discussed.