The nature of turbulence in compressible homentropic constant shear flows: its vortex and wave contents and self-sustenance.

本项目的目的是研究谱稳定的可压缩均匀剪切流中湍流的维持机制。我们提出该提案的动机是最近在研究不可压缩和可压缩剪切流湍流动力学方面取得的进展（G。Mamatsashvili等人，“均质剪切湍流的动力学：非线性横向叶栅在旁路概念中的关键作用”，《物理学评论》E，94，2016年和Hau等人，等熵恒定剪切流中涡旋扰动产生的线性和非线性空气动力学声音的比较数值分析，流体物理，27（2015））。在那里，我们研究了傅立叶谐波和非线性过程的线性瞬态增长的相互作用。在这个谱稳定流的线性增长的谐波具有瞬态性质，是强烈的各向异性的频谱空间。这反过来又导致频谱空间中的非线性过程的各向异性，因此，主要的非线性过程似乎不是直接/反向的，而是傅立叶空间中谐波的横向/角度重新分布，称为非线性横向级联。本文针对不可压均质剪切流，对这种新型的非线性横向叶栅进行了详细的研究和分析。我们证明，湍流是由线性瞬态增长和非线性横向级联的相互作用维持的。此外，还表明，在这些类型的流动中的湍流可以用可压缩涡模和声波来描述。开发了一种分离这些模式的精细程序，这也将是本项目使用的基本方法之一。所产生的声场在波数平面中是各向异性的，这导致高度定向的线性声辐射，而非线性产生的波几乎是全向的。它的来源是由非正规性引起的线性模式耦合，这在中等马赫数下变得有效。在可压缩均匀剪切流中，涡和声波模式线性耦合。这导致不可避免地从涡流模式中产生声波模式，并可能与横向叶栅相连。因此，出于这些新奇的动机，我们建议进行可压缩均匀剪切流的谱空间中的湍流动力学分析，以研究非线性横向级联如何在那里表现出来，因为固有的可压缩性效应可能会发挥作用，影响动力学。这将通过模拟梯度和湍流马赫数变化的均匀剪切湍流来实现。

he aim of this project is to investigate the mechanism of sustenance of turbulence in spectrally stable compressible homogeneous shear flow. The motivation of our proposal is the progress achieved recently when studying the dynamics of incompressible and compressible shear flow turbulence (G. Mamatsashvili et al., “Dynamics of homogeneous shear turbulence: A key role of the nonlinear transverse cascade in the bypass concept”, Phys.Rev.E, 94, 2016 and Hau et al., A comparative numerical analysis of linear and nonlinear aerodynamic sound generation by vortex disturbances in homentropic constant shear flows, Physics of Fluids, 27 (2015)). There we examined the interplay of linear transient growth of Fourier harmonics and nonlinear processes. In this spectrally stable flow the linear growth of the harmonics has a transient nature and is strongly anisotropic in spectral space. This, in turn, leads to anisotropy of nonlinear processes in spectral space and, as a result, the main nonlinear process appears to be not a direct/inverse, but rather a transverse/angular redistribution of harmonics in Fourier space referred to as the nonlinear transverse cascade. In our paper, this new nonlinear transverse cascade was studied and analysed in detail for incompressible homogeneous shear flow. We demonstrated, that the turbulence is sustained by the interplay of the linear transient growth and the nonlinear transverse cascade. It was shown additionally, that turbulence in these type of flows be described by compressible vortex modes and acoustic waves. A refined procedure of separation of these modes was developed, which will be one of the basic methodologies used for this project, too. The generated acoustic field is anisotropic in the wavenumber plane, which results in highly directional linear sound radiation, whereas the nonlinearly generated waves are almost omni-directional. Its source is the linear mode-coupling induced by non-normality, which becomes efficient at moderate Mach numbers. In compressible homogeneous shear flows vortex and acoustic wave modes are linearly coupled. This leads to the inevitable generation of acoustic wave modes from the vortex ones and a likely connection to the transverse cascade. Thus, motivated by these novelties, we propose to perform the analysis of the turbulence dynamics in spectral space for compressible homogeneous shear flow to investigate how the nonlinear transverse cascade manifests itself there, as intrinsic compressibility effects could come into play, influencing the dynamics. This will be achieved by simulating homogeneous shear turbulence subject to varying gradient and turbulent Mach numbers.