Structure and stability of non-symmetric Burgers vortices
Structure and stability of non-symmetric Burgers vortices
复制标题
非对称伯格斯涡的结构与稳定性
DOI：
10.1017/s0022112098008866
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发表时间：
1998
影响因子：
3.7
通讯作者：
D. Pullin
中科院分区：
文献类型：
--
作者：
A. Prochazka;D. Pullin
我们通过数值和分析研究了纳维-斯托克斯方程的稳态和准稳态解的结构和稳定性，该方程对应于嵌入均匀非对称应变场 (αx, βy, γz)、α+β+γ=0 中的拉伸涡旋，其中拉伸应变的一个主轴与涡度对齐。这些被称为非对称伯格涡旋（Robinson & Saffman 1984）。我们考虑涡旋雷诺数 R=Г/(2πv)，其中 Г 为涡旋环流，v 为运动粘度，范围为 R=1−104，应变比范围广泛 λ=(β−α)/(β+α)，包括 λ>1，在某些情况下为 λ[Gt ]1。伪谱方法用于获得与包括 λ 在内的整个 (R, λ) 参数空间上的稳态和准稳态涡旋状态相对应的数值解，其中 Moffatt、Kida 和 Ohkitani (1994) 提出的论点证明了严格稳态解的不存在。当 λ[Gt ]1、R[Gt ]1 和 ε≡λ/R[Lt ]1 时，我们在 1<r/(2v/γ)1/2[les ]ε1/2 区域找到了涡量的精确渐近形式，与我们的数值解非常吻合。这表明存在一个扩展区域，其中指数小涡度被限制在几乎二维流的近猫眼形区域内，并且在边界流线上取几乎等于 γ/(4πv)exp[−1/(2eε)] 的常数值。这允许将循环的泄漏率估计为无穷大，如 ∂Г/∂t =(0.48475/4π)γε−1Г exp (−1/2eε)，当 λ>1 时，涡流相应呈指数缓慢衰减。基于幂法的迭代技术用于估计非对称情况 λ>0 的最大特征值。发现了 0[les ]λ[les ]1 的稳定性，并且发现并分析了 λ>1 的中性对流不稳定模式。我们的一般结论是，广义非对称伯格涡旋对于所有 R 0[les ]λ[les ]1 的二维扰动都是无条件稳定的，并且当 λ>1 时，涡旋只会通过指数缓慢的涡度泄漏而衰减，这表明在这种情况下具有极高的鲁棒性。
We investigate, numerically and analytically, the structure and stability of steady and quasi-steady solutions of the Navier–Stokes equations corresponding to stretched vortices embedded in a uniform non-symmetric straining field, (αx, βy, γz), α+β+γ=0, one principal axis of extensional strain of which is aligned with the vorticity. These are known as non-symmetric Burgers vortices (Robinson & Saffman 1984). We consider vortex Reynolds numbers R=Γ/(2πv) where Γ is the vortex circulation and v the kinematic viscosity, in the range R=1−104, and a broad range of strain ratios λ=(β−α)/(β+α) including λ>1, and in some cases λ[Gt ]1. A pseudo-spectral method is used to obtain numerical solutions corresponding to steady and quasi-steady vortex states over our whole (R, λ) parameter space including λ where arguments proposed by Moffatt, Kida & Ohkitani (1994) demonstrate the non-existence of strictly steady solutions. When λ[Gt ]1, R[Gt ]1 and ε≡λ/R[Lt ]1, we find an accurate asymptotic form for the vorticity in a region 1<r/(2v/γ)1/2[les ]ε1/2, giving very good agreement with our numerical solutions. This suggests the existence of an extended region where the exponentially small vorticity is confined to a nearly cat's-eye-shaped region of the almost two-dimensional flow, and takes a constant value nearly equal to Γγ/(4πv)exp[−1/(2eε)] on bounding streamlines. This allows an estimate of the leakage rate of circulation to infinity as ∂Γ/∂t =(0.48475/4π)γε−1Γ exp (−1/2eε) with corresponding exponentially slow decay of the vortex when λ>1. An iterative technique based on the power method is used to estimate the largest eigenvalues for the non-symmetric case λ>0. Stability is found for 0[les ]λ[les ]1, and a neutrally convective mode of instability is found and analysed for λ>1. Our general conclusion is that the generalized non-symmetric Burgers vortex is unconditionally stable to two-dimensional disturbances for all R, 0[les ]λ[les ]1, and that when λ>1, the vortex will decay only through exponentially slow leakage of vorticity, indicating extreme robustness in this case.