A new formulation for the 3-D Euler equations with an application to subsonic flows in a cylinder

TL;DR

This paper introduces a novel formulation of the 3-D Euler equations that decouples hyperbolic and elliptic modes, enabling the construction of smooth subsonic flows in a cylinder with new conservation laws.

Contribution

The paper develops a new formulation of the 3-D Euler equations using Bernoulli's law, leading to novel conservation laws and facilitating flow construction in cylindrical geometries.

Findings

Derived a decoupling of hyperbolic and elliptic modes in 3-D Euler equations.

Identified a conserved quantity similar to scaled vorticity for flows with constant Bernoulli's function.

Constructed smooth subsonic Euler flows in a rectangular cylinder.

Abstract

In this paper, a new formulation for the three dimensional Euler equations is derived. Since the Euler system is hyperbolic-elliptic coupled in a subsonic region, so an effective decoupling of the hyperbolic and elliptic modes is essential for any development of the theory. The key idea in our formulation is to use the Bernoulli's law to reduce the dimension of the velocity field by defining new variables $(1,\beta_2=\frac{u_2}{u_1},\beta_3=\frac{u_3}{u_1})$ and replacing $u_1$ by the Bernoulli's function $B$ through $u_1^2=\frac{2(B-h(\rho))}{1+\beta_2^2+\beta_3^2}$. We find a conserved quantity for flows with a constant Bernoulli's function, which behaves like the scaled vorticity in the 2-D case. More surprisingly, a system of new conservation laws can be derived, which is new even in the two dimensional case. We use this new formulation to construct a smooth subsonic Euler flow in a rectangular cylinder, which is also required to be adjacent to some special subsonic states.