Title: On the Interaction between Turbulence and a Planar Rarefaction Show Chinese title

Title: 关于湍流与平面稀疏之间的相互作用

Abstract: The modeling of turbulence, whether it be numerical or analytical, is a difficult challenge. Turbulence is amenable to analysis with linear theory if it is subject to rapid distortions, i.e., motions occurring on a time scale that is short compared to the time scale for non-linear interactions. Such an approach could prove useful for understanding aspects of astrophysical turbulence, which is often subject to rapid distortions, such as supernova explosions or the free-fall associated with gravitational instability. As a proof of principle, a particularly simple problem is considered here: the evolution of vorticity due to a planar rarefaction in an ideal gas. Vorticity can either grow or decay in the wake of a rarefaction front, and there are two competing effects that determine which outcome occurs: entropy fluctuations couple to the mean pressure gradient to produce vorticity via baroclinic effects, whereas vorticity is damped due to the conservation of angular momentum as the fluid expands. In the limit of purely entropic fluctuations in the ambient fluid, a strong rarefaction generates vorticity with a turbulent Mach number on the order of the root-mean square of the ambient entropy fluctuations. The analytical results are shown to compare well with results from two- and three-dimensional numerical simulations. Analytical solutions are also derived in the linear regime of Reynolds-averaged turbulence models. This highlights an inconsistency in standard turbulence models that prevents them from accurately capturing the physics of rarefaction-turbulence interaction. Finally, dimensional analysis of the equations indicates that rapid distortion of turbulence can give rise to two distinct regimes in the turbulent spectrum: a distortion range at large scales where linear distortion effects dominate, and an inertial range at small scales where non-linear effects dominate. Show Chinese abstract

Abstract: 湍流的建模，无论是数值的还是分析的，都是一个困难的挑战。 如果湍流受到快速变形的影响，即在与非线性相互作用的时间尺度相比很短的时间尺度上发生运动，那么湍流可以通过线性理论进行分析。 这种方法可能有助于理解天体物理湍流的某些方面，因为天体物理湍流通常受到快速变形的影响，例如超新星爆炸或与重力不稳定性相关的自由下落。 作为原理证明，这里考虑了一个特别简单的问题：理想气体中平面稀疏波引起的涡度演化。 涡度可以在稀疏波前沿增长或衰减，有两个竞争效应决定了哪种结果会发生：熵波动通过压强梯度产生涡度，这是由于巴罗克拉因效应，而涡度则由于流体膨胀时角动量守恒而被阻尼。 在环境流体中纯熵波动的极限情况下，强烈的稀疏波会产生具有湍流马赫数的涡度，其数量级约为环境熵波动的均方根值。 分析结果与二维和三维数值模拟的结果显示出良好的一致性。 还推导出了雷诺平均湍流模型线性区域的解析解。 这突显了标准湍流模型中的不一致之处，阻碍了它们准确捕捉稀疏波-湍流相互作用的物理机制。 最后，对方程的量纲分析表明，湍流的快速变形可以导致湍流谱中的两个不同区域：在大尺度上的变形区，其中线性变形效应占主导地位；在小尺度上的惯性区，其中非线性效应占主导地位。