螺旋湍流指的是局部或平均螺旋度不为零的湍流流动状态，广泛存在于航空发动机内流、高超声速飞行器、惯性约束核聚变以及星系密度云演化等工程及自然现象过程中。上世纪六十年代螺旋度守恒性定理的发现，给湍流的理论研究提供了新的方向，几十年来主要在不可压缩均匀各向同性湍流等领域取得系列研究进展。相对于常规湍流而言，螺旋湍流有许多独特之处，如经典的-5/3标度发生改变，方程的非线性特征被弱化，飞行器气动阻力减小，发动机燃料混合效率提高等。近些年来，面向航空航天及聚变领域对湍流基础研究的重大需求，结合螺旋度对湍流性质的影响规律，有必要发展适用于可压缩、各向异性条件下的螺旋湍流理论。

   本文主要开展螺旋湍流的机理研究，采用理论推导及数值模拟的方法，重点关注可压缩及各向异性条件下相关螺旋湍流理论的研究，一方面是探索螺旋湍流内部物理过程，揭示湍流发生及发展规律，另一方面是为湍流建模，尤其是大涡模拟建模提供理论指导。为有效支撑所建立的湍流理论，采用直接数值模拟方法，建立了不可压缩均匀各向同性湍流、可压缩均匀各向同性湍流及流向旋转槽道湍流数据库。论文的创新点主要包括以下三个方面：

(一).提出了螺旋度级串的双通道及尺度局部性理论

    为了描述一般各向异性流动中螺旋度跨尺度传输的特征，在传统螺旋度级串单通道的基础上，本文提出了螺旋度级串双通道的概念，并通过理论分析方法给出了双通道的理论表达式。经过严密的数学推导，证实所提出的螺旋度级串双通道在三维湍流中三个方向同时满足均匀性条件下，只考虑系综平均的层次上是完全等效的。因此，第二通道的提出，可弥补均匀湍流中螺旋度级串的高阶统计矩问题，同时可适用于非均匀湍流中螺旋度级串的研究。第二通道的提出，将二阶反对称张量引入到湍流级串理论研究中来，以矩阵张量几何为视角，研究湍流级串涉及的矩阵数学性质问题。

    结合上述提出的螺旋度双通道理论，我们进一步在物理空间研究螺旋度级串的尺度局部性问题。尺度局部性是经典湍流级串理论的基本假设，可证实湍流的小尺度统计特征具有普适性。而在物理空间研究湍流级串问题，需要引入准正则假设，具有一定的任意性。引入带宽滤波的分析方法，提取流场中两个特定的尺度，来研究特定尺度间的能量及螺旋度传输规律。数据库分析结果表明，能量及螺旋度的正级串和反级串均满足尺度局部性的特征，并定量给出正反级串的尺度局部性范围。

(二).建立了可压缩螺旋湍流的联合级串理论

    经典的湍流级串理论建立在动能及螺旋度是二次无粘守恒量的基础上，然而，在可压缩湍流中，动能及螺旋度的这种守恒性特征被压力破坏，成为将经典湍流理论向可压缩状态下推广的主要障碍。通过压力涡量密度协同谱分析方法及湍流数据库的验证，可以证实压力在统计意义上只在大尺度范围内起作用。意味着超出一定的临界尺度，动能及螺旋度可恢复其守恒性的特征，这个大尺度范围一般指含能区范围。另外，我们发现，动能及螺旋度级串对应的临界尺度有明显的差异，螺旋度级串的临界尺度更大，意味着螺旋度级串惯性区的上临界范围更大，证实前期基于螺旋度所建立的大涡模拟模型的网格适用范围更广。

(三).给出了可压缩湍流中三波交互涉及的手性传输规律

    可压缩湍流的三波交互问题，不仅涉及速度场胀压模态的引入，还涉及流场矢量与热力学量的耦合。本文以手性分解为视角，探索可压缩湍流中三波交互涉及的手性传输问题，重点关注压缩膨胀过程对手性传输的影响规律。拓展了经典螺旋波分解特征值的内涵，将亥姆霍兹分解和螺旋波分解统一起来，提出了广义螺旋波分解方法，为流场速度矢量提供了三个两两相互作用正交的基向量。研究结果表明，速度场的胀压模态对手性传输过程扮演着媒介的作用，对手性间的三波交互过程有着重要的影响。发现压缩促进手性平衡，而膨胀则促进手性非平衡，使得流场的螺旋度效应更加明显。

Turbulent flow with non-zero helicity is called helical turbulence, which appears widely in engineering and natural phenomena, such as aero-engine internal flows, hypersonic aircraft, inertial confinement fusion, and galactic density cloud evolutions. The discovery of the conservation theorem of helicity in the 1960s provided a new direction for the theoretical study of turbulence. In the past few decades, a series of research progress has been made in the area of incompressible homogeneous and isotropic turbulence. Compared with the common turbulent flows, there exist many unique features in helical turbulent flows, such as the change of the classical -5/3 scaling, the weakening of the nonlinear characteristics of the governing equations, the reduction of the aerodynamic drag of the aircraft, and the improvement of the fuel mixing efficiency of the engine, etc. Basing on the demand of the turbulence theory in the fields of aerospace and fusion, it is necessary to consider the compressibility and anisotropy effect to develop helical turbulence theory.

  We adopt the methodology of theoretical derivation and numerical simulations, and explore the statistical theory of helical turbulence with compressibility and anisotropy. On the one hand is to investigate the intrinsic physical process, revealing the regulations of transitions and developments of helical turbulence. On the other hand is to provide theoretical suggestions for turbulence modelings, especially large eddy simulations. We carry out direct numerical simulations to build datasets of incompressible homogeneous and isotropic turbulence, compressible homogeneous and isotropic turbulence, and turbulent channel flows with streamwise rotation. The main innovation points can be summarized as the following three aspects:

  Firstly, we propose the theories of dual channels and scale locality of helicity cascade. In order to describe the characteristics of cross-scale helicity transfer in general anisotropic flows, the dual channels of helicity cascade are developed based on the traditional single channel of helicity cascade, and we deduce their theoretical expressions. Through rigorous mathematical derivation, it is proved that the newly proposed channel is completely equivalent to the traditional channel when only considering the ensemble average with three-dimensional homogeneity. Hence, the dual channels can describe the high-order statistical moments in homogeneous turbulence and be suitable for inhomogeneous turbulence. In addition, the second-order antisymmetric tensor is introduced into the research of turbulent cascade theory. From the perspective of tensor geometry, we study the mathematics of turbulent cascade.

  Based on the above methodology, we further study the scale locality of helicity cascade in physical space. Scale locality is a basic assumption of the classical turbulent cascade theory, which proves that the small-scale statistical characteristics of turbulence are universal. The quasi-normal-type scheme is needed to study turbulence cascade in physical space, which is arbitrary to some extent. The bandwidth filtering method is induced to extract two specific scales to study the cross-scale transfer of kinetic energy and helicity. The statistical results indicate that the forward and backward cascade of kinetic energy and helicity satisfy the characteristics of scale locality, and the scale locality ranges are given quantitatively.

  Secondly, we build the joint cascade theory in compressible helical turbulence. The classical turbulence cascade theory is based on the fact that kinetic energy and helicity are quadratic inviscid invariants. However, their conservation characteristics are destroyed by pressure terms, which becomes a major obstacle time the generalization of classical turbulence theory to compressible turbulence. We conclude that pressure terms only play a role in the large scales in the statistical sense, through the analysis of the pressure vorticity cospectrum and turbulence dataset verification. It means that beyond a certain critical scale, the conservation characteristics of kinetic energy and helicity can be recovered, and these large-scale ranges refer to the energy-containing regions. In addition, the critical scales of kinetic energy and helicity are different, and the critical scale of helicity cascade is larger. This phenomenon means that the previous turbulence modelings based on the helicity are valid in coarser meshes.

  Thirdly, we explore the chirality transfer regulations involved in the triple interactions in compressible turbulence. The compressible mode of velocity field and the couplings of vectors and thermodynamics complex the triple interactions of compressible turbulence. From the perspective of chiral decompositions, we focus on the compressibility effects for the chirality transfer of triple interactions. We extend the connotation of the characteristic values of the classical helical wave decomposition, and propose a general helical wave decomposition method to providing three orthogonal basis vectors. We found that the compressible mode of velocity field acts as a medium for chirality transfer, and has an important influence on the triple interactions. It is found that compression promotes chiral equilibrium, while expansion promotes chiral nonequilibrium, which highlights the helicity effects in turbulent flows.