Angular dependence and growth of vorticity in the three-dimensional Euler equations

Abstract

An investigation of lower bounds on the quantities Wn(t)=∫Ω|ω|2n dV for n⩾1 for the incompressible three-dimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted “eigenvalues,” λS(n)(t) and λP(n)(t), of the strain matrix S and the Hessian matrix of the pressure P={p,ij}, respectively. It is shown that these obey the simple inequality, λ̇S(n)+f(θn)(λS(n))2+λP(n)⩾0, where f(θn)=1−tan2 θn. The θn are spatially averaged weighted angles between the vorticity vector ω and the vortex stretching vector σ=ω⋅∇u. The weighting in the averaging process highlights regions of large vorticity. This is the angle considered by Tsinober, Kit, and Dracos in their analysis of data from turbulent grid flow experiments in which they noted a tendency toward alignment between ω and σ. The Burgers vortex turns out to be a sharp solution of this inequality with a corresponding angle θn=0, giving rise to exponential growth in Wn. Some special solutions for cases where θn moves between θn=0 and θn=π/2 are displayed. The work of Ohkitani and Kishiba on the alignment in 3-D Euler flows between ω and the third eigenvector of P at maximum enstrophy is also particularly relevant and is applied to the modified pressure matrix Q={p,ij−3δijp,ii} in the limit n→∞. The finite time blow-up problem is discussed in this context. In an Appendix it is shown that an identical inequality holds for the barotropic compressible Euler equations where ζ=ω/ρ and Wn(t)=∫Ωρ|ζ|2n dV replace ω and Wn, respectively.