New solution of the compressible Navier-Stokes equation

TL;DR

This paper presents a new smooth solution to the compressible Navier-Stokes equation that remains valid at all times, derived from the exact solution of the Euler vortex equation, and connects to known energy spectra in 2D flows.

Contribution

The authors derive a novel smooth solution to the compressible Navier-Stokes equation with pressure proportional to divergence, extending previous solutions and linking to classical energy spectra.

Findings

New smooth solution to compressible Navier-Stokes equations

Exact representation of 2D energy spectrum

Connection to known turbulence spectra

Abstract

We use the general exact solution of the Cauchy problem for the compressible Euler vortex equation in unbounded space which was obtained earlier (S.G.Chefranov, Sov. Phys. Dokl., 36, 286, 1991). This solution loses its smoothness in finite time and coincides with the exact solution of the Hopf equation, describing the inertial motion of the ideal fluid without pressure. On this base we obtain here the new smooth at all times solution to the compressible Navier-Stokes (NS) equation with the pressure field shows linear proportionality to the divergence of the velocity field, as it is known for an out-of-equilibrium systems with large second viscosity and small first viscosity. For example, directly from this solution of the NS equation for the case of two-dimensional (2D) compressible flow the exact representation of energy spectrum well known for 2D incompressible case (R.H.Kraichnan, Phys.Fluids,vol.10,1417,1967) is obtained.