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可压缩均匀剪切流中湍流的维持机制。我们提议的动机是最近在研究不可压缩和可压缩剪切流动湍流动力学方面取得的进展(G.Mamatsashvili等人，“均匀剪切湍流的动力学：非线性横向叶栅在旁路概念中的关键作用”，Phys.Rev.E，94,2016和Hau等人，“同质恒切变流中涡流扰动产生线性和非线性气动声音的比较数值分析”，流体物理学，27(2015))。在那里，我们研究了傅里叶调和的线性瞬时增长和非线性过程的相互作用。在这种频谱稳定的流动中，谐波的线性增长具有瞬变性质，并且在频谱空间中具有强烈的各向异性。这又导致了频谱空间中非线性过程的各向异性，因此，主要的非线性过程似乎不是正/逆，而是傅里叶空间中谐波的横向/角度重分布，称为非线性横向级联。本文对这种新的非线性横向叶栅在不可压缩均匀剪切流动中进行了详细的研究和分析。我们证明，湍流是由线性瞬变增长和非线性横向叶栅相互作用所支撑的。研究还表明，这类流动中的湍流可用可压缩涡模和声波来描述。开发了一种分离这些模式的改进程序，这也将是本项目使用的基本方法之一。产生的声场在波数平面上是各向异性的，这导致了高方向性的线性声辐射，而非线性产生的波几乎是全向的。它的源是由非正态引起的线性模耦合，这种耦合在中等马赫数时变得有效。在可压缩均质剪切流中，涡旋和声波模式是线性耦合的。这导致了涡旋模式不可避免地产生声波模式，并可能与横向叶栅连接。因此，受这些新颖性的启发，我们建议在谱空间中对可压缩均匀剪切流的湍流动力学进行分析，以研究非线性横向叶栅是如何在谱空间中表现出来的，因为内在的可压缩性效应可以发挥作用，影响动力学。这将通过模拟具有不同梯度和湍流马赫数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.