涡量和应变的近相消需要剪切或平衡涡旋

中文摘要 AI 辅助

在压力泊松方程的非线性源项中，脉动涡量和应变几乎相互抵消。我们探究何处抵消是完全的（称之为“静默”），以及源项如何逃脱这种抵消。逐点地，源项仅依赖于速度梯度的特征值。纯剪切没有特征值，因此它是静默的。它是归一化不变量平面的原点，其第二不变量坐标即为归一化源项。设$f$为梯度的剪切分数，$m$为其到该原点的距离。我们证明在任意不可压缩流动的每一点上，$m\le1-f\le3m$。凡源项几乎消失且第三不变量很小之处，梯度近似为纯剪切。否则，抵消梯度为平衡涡旋，即一种旋转，其自身的应变在任何强度下都能抵消。在无滑移壁面处，静默对象是精确的：壁面剪应力脉动，即秩一薄片。远离壁面时，我们假设同样的薄片，附有小的残余，并推导其如何逃脱。逃脱过程经过残余的一个元素，即沿薄片方向的薄片法向速度变化。其振幅为薄片与残余的几何平均，其符号选择涡旋或应变。秩二静默对象具有相同的源项，通过相同的通道，但特征值平衡，因此该假设确定了逃脱的拓扑结构，而非其源项。因此，剪切分数是剪切抵消的逐点代理，源项模型必须建立在逃脱机制之上。

英文摘要

In the nonlinear source of the pressure Poisson equation the fluctuating enstrophy and strain nearly cancel. We ask where the cancellation is complete, which we call silent, and how the source escapes it. Pointwise the source depends only on the eigenvalues of the velocity gradient. Pure shear has none, so it is silent. It is the origin of the normalised invariant plane, whose second-invariant coordinate is the normalised source. Let $f$ be the shear fraction of the gradient and $m$ its distance from that origin. We prove that $m\le1-f\le3m$ at every point of every incompressible flow. Wherever the source nearly vanishes and the third invariant is small, the gradient is nearly pure shear. Otherwise the cancelling gradient is a balanced vortex, a swirl its own strain cancels at any strength. At a no-slip wall the silent object is exact: the wall shear-stress fluctuation, a rank-one sheet. Away from the wall we assume the same sheet, dressed with a small residual, and derive how it escapes. The escape passes through one element of the residual, the along-sheet variation of the sheet-normal velocity. Its amplitude is the geometric mean of sheet and residual, and its sign selects swirl or strain. A rank-two silent object has the same source through the same gates but balanced eigenvalues, so the assumption fixes the escape's topology, not its source. The shear fraction is thus a pointwise proxy for cancellation by shear, and a model of the source must be built on what escapes.

发表机构

• Center for Turbulence Research, Stanford University（斯坦福大学湍流研究中心）

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