Stability results for idealized shear flows on a rectangular periodic domain
HR Dullin, J Worthington - Journal of Mathematical Fluid Mechanics, 2018 - Springer
Journal of Mathematical Fluid Mechanics, 2018•Springer
We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-
dimensional rectangular periodic domain 0, 2 π) * 0, 2 π/κ) 0, 2 π)× 0, 2 π/κ) for κ ∈ R^+ κ∈
R+, the Euler equations admit a family of stationary solutions given by the vorticity profiles
Ω^*(x)= Γ\cos (p_1x_1+ κ p_2x_2) Ω∗(x)= Γ cos (p 1 x 1+ κ p 2 x 2). We show linear stability
for such flows when p_2= 0 p 2= 0 and κ ≥| p_1| κ≥| p 1|(equivalently p_1= 0 p 1= 0 and κ|
p_2| ≤ 1 κ| p 2|≤ 1). The classical result due to Arnold is that for p_1= 1, p_2= 0 p 1= 1, p 2 …
dimensional rectangular periodic domain 0, 2 π) * 0, 2 π/κ) 0, 2 π)× 0, 2 π/κ) for κ ∈ R^+ κ∈
R+, the Euler equations admit a family of stationary solutions given by the vorticity profiles
Ω^*(x)= Γ\cos (p_1x_1+ κ p_2x_2) Ω∗(x)= Γ cos (p 1 x 1+ κ p 2 x 2). We show linear stability
for such flows when p_2= 0 p 2= 0 and κ ≥| p_1| κ≥| p 1|(equivalently p_1= 0 p 1= 0 and κ|
p_2| ≤ 1 κ| p 2|≤ 1). The classical result due to Arnold is that for p_1= 1, p_2= 0 p 1= 1, p 2 …
Abstract
We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain for , the Euler equations admit a family of stationary solutions given by the vorticity profiles . We show linear stability for such flows when and (equivalently and ). The classical result due to Arnold is that for and the stationary flow is nonlinearly stable via the energy-Casimir method. We show that for the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all steady states satisfying . The modification and application of a structure-preserving Hamiltonian truncation is discussed for the anisotropic case . This leads to an explicit Lie-Poisson integrator for the approximate system, which is used to illustrate our analytical results.
Springer
以上显示的是最相近的搜索结果。 查看全部搜索结果