Stability Results for Idealized Shear Flows on a Rectangular Periodic Domain
Stability Results for Idealized Shear Flows on a Rectangular Periodic Domain
复制标题
矩形周期域上理想化剪切流的稳定性结果
DOI：
10.1007/s00021-017-0329-2
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发表时间：
2016
影响因子：
1.3
通讯作者：
J. Worthington
中科院分区：
文献类型：
--
作者：
H. Dullin;J. Worthington
本文给出了环面上欧拉流体流动的一个新的线性稳定解。在二维矩形周期域$$[0，2\pi）\times [0，2\pi/ \kappa）$$[0，2 π）×[0，2 π/κ）上，对于$$\kappa \in \mathbb {R}^+$$κ∈R+，欧拉方程组承认由涡量分布图$\Ω ^* 给出的一族定常解（\mathbf {x}）= \Gamma \cos（p_1x_1+ \kappa p_2x_2）$$Ω（x）=Γcos（p1x1+κp2x2）.当$$p_2= 0 $$p2 =0和$$\kappa \ge时，我们证明了这类流的线性稳定性|p_1| $$κ≥| P1|（相当于$$p_1=0$$p1=0和$$\kappa {|p_2|}\le {1}$$κ| P2| ≤1）。Arnold的经典结果是：当p1 = 1，p2 = 0，p1 =1，p2=0且$\kappa \ge 1$$κ≥1时，通过能量-Casimir方法，定常流是非线性稳定的.我们证明对于$$\kappa \ge| p_1|\ge 2，p_2 = 0$$κ≥| P1|当p2=0时，流动是线性稳定的，但不能期望类似的非线性稳定性结果。最后，我们证明了满足$$p_1^2+\kappa ^2{p_2^2}>\frac{{3（\kappa ^2+1）}}{4（7-4\sqrt{3}）}$$p12+κ 2 p22>3（κ2+1）4（7-43）的所有定态的非线性不稳定性。讨论了各向异性情形$$\kappa \ne 1$$κ 1的保结构哈密顿截断的修正和应用。这导致一个明确的李泊松积分的近似系统，这是用来说明我们的分析结果。
We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $$[0,2\pi )\times [0,2\pi / \kappa )$$[0,2π)×[0,2π/κ) for $$\kappa \in \mathbb {R}^+$$κ∈R+, the Euler equations admit a family of stationary solutions given by the vorticity profiles $$\Omega ^*(\mathbf {x})= \Gamma \cos (p_1x_1+ \kappa p_2x_2)$$Ω∗(x)=Γcos(p1x1+κp2x2). We show linear stability for such flows when $$p_2=0$$p2=0 and $$\kappa \ge |p_1|$$κ≥|p1| (equivalently $$p_1=0$$p1=0 and $$\kappa {|p_2|}\le {1}$$κ|p2|≤1). The classical result due to Arnold is that for $$p_1 = 1, p_2 = 0$$p1=1,p2=0 and $$\kappa \ge 1$$κ≥1 the stationary flow is nonlinearly stable via the energy-Casimir method. We show that for $$\kappa \ge |p_1| \ge 2, p_2 = 0$$κ≥|p1|≥2,p2=0 the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all steady states satisfying $$p_1^2+\kappa ^2{p_2^2}>\frac{{3(\kappa ^2+1)}}{4(7-4\sqrt{3})}$$p12+κ2p22>3(κ2+1)4(7-43). The modification and application of a structure-preserving Hamiltonian truncation is discussed for the anisotropic case $$\kappa \ne 1$$κ≠1. This leads to an explicit Lie-Poisson integrator for the approximate system, which is used to illustrate our analytical results.