Linear stability analysis of parallel shear flows for an inviscid generalized two-dimensional fluid system
Linear stability analysis of parallel shear flows for an inviscid generalized two-dimensional fluid system
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DOI：
10.1088/1751-8113/46/6/065501
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发表时间：
2013-02
期刊：
影响因子：
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通讯作者：
T. Iwayama;Masakazu Sueyoshi;Takeshi Watanabe
中科院分区：
文献类型：
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作者：
T. Iwayama;Masakazu Sueyoshi;Takeshi Watanabe
研究了无粘广义二维流体系统（即α湍流系统）平行剪切流的线性稳定性。该系统的特征为平流标量q与流函数ψ之间的关系式q =−（−Δ）α/2ψ。这里，α是一个不超过3的实数，q称为广义涡量。本文利用波浪活度守恒导出了平行剪切流线性稳定的充分条件。然后对违反稳定条件的片状涡进行了稳定性分析。二维欧拉系统（α = 2）中片状涡的不稳定性称为开尔文-亥姆霍兹（KH）不稳定性；研究了广义二维流体系统在0 < α < 3时的不稳定性。片状涡旋是不稳定的，因为施加于它的正弦扰动随时间呈指数增长。增长率是有限的，取决于扰动的波数k3−α对于1 < α < 3，其中k是扰动的波数。相反，当0 < α≤1时，生长速率是无限大的。换句话说，摄动的增长率在α = 1处发生转变。提出了广义二维流体系统KH不稳定性的物理模型，该模型可以解释微扰增长率在α = 1时的转变。
The linear stability of parallel shear flows for an inviscid generalized two-dimensional (2D) fluid system, the so-called α turbulence system, is studied. This system is characterized by the relation q = −( − Δ)α/2ψ between the advected scalar q and the stream function ψ. Here, α is a real number not exceeding 3 and q is referred to as the generalized vorticity. In this study, a sufficient condition for linear stability of parallel shear flows is derived using the conservation of wave activity. A stability analysis is then performed for a sheet vortex that violates the stability condition. The instability of a sheet vortex in the 2D Euler system (α = 2) is referred to as a Kelvin–Helmholtz (KH) instability; such an instability for the generalized 2D fluid system is investigated for 0 < α < 3. The sheet vortex is unstable in the sense that a sinusoidal perturbation applied to it grows exponentially with time. The growth rate is finite and depends on the wavenumber of the perturbation as k3 − α for 1 < α < 3, where k is the wavenumber of the perturbation. In contrast, for 0 < α ⩽ 1, the growth rate is infinite. In other words, a transition of the growth rate of the perturbation occurs at α = 1. A physical model for KH instability in the generalized 2D fluid system, which can explain the transition of the growth rate of the perturbation at α = 1, is proposed.