Transition to turbulence and unstable nonlinear magnetohydrodynamic flow states in ducts with a transverse magnetic field

在各种流动系统中，向湍流的转变是重要的。当雷诺数Re增加时，它通常在壁边界流中发生亚临界，即，与层流状态的不稳定性无关。这种转变通常在动力系统理论的框架中解释。 流动状态被认为是可能的流动状态的相空间中的一个点。层流和湍流具有不同的吸引域，它们被超曲面（混沌边缘）分开。在这个表面上，流动既不朝向层流状态发展，也不朝向湍流状态发展，而是朝向表面内的吸引子发展，该吸引子被称为边缘状态。在过渡到湍流的过程中，流动可能首先到达边缘状态附近，然后才最终变成湍流。边缘状态对应于非平凡的时间依赖流，其动力学比湍流解更简单。因此，它们可以作为研究转变的物理机制的模型。边缘状态可以，例如，对应于波动、时间周期或混沌解。他们详细研究了一些典型的剪切流，但几乎从来没有被认为是磁流体动力学（MHD）流，虽然在这些流的过渡也是亚临界的。在MHD流动中，哈特曼数Ha作为磁场的第二个参数出现，除了雷诺数作为速度的无量纲参数。在该项目中，将确定在很大程度上未知的MHD管道流中的湍流路径和转捩机制。研究的具体结构是矩形截面和横向均匀磁场管道中的流动。这些流动是三维的，其特征是在平行和垂直于磁场的壁上的边界层，分别称为Shercliff层和Hartmann层。将通过高分辨率直接数值模拟（DNS）和稳定性分析来研究流动。首先，在Re-Ha-A参数空间中确定层流和湍流的边界，其中A是初始状态的扰动幅度。在第二步中，边缘状态将由DNS确定，并将分析它们的动力学和特定属性。在第三步中，流动状态的分叉时的Re和Ha的变化将被调查，产生的边缘状态。通过这一点，人们也可以获得更好的理解，为什么过渡是由一个参数，这是基于哈特曼层的厚度，虽然湍流首先出现在Shercliff层。我们的结果与导电液体冶金和传热应用中的电磁流量控制相关。

Transition to turbulence is important in various flow systems. Upon increasing the Reynolds number Re, it typically occurs sub-critically in wall-bounded flows, i.e. irrespective of an instability of the laminar flow state. The transition is typically interpreted in the framework of dynamical systems theory. A flow state is regarded as a point in the phase space of possible flow states. Laminar and turbulent flow have different basins of attraction that are separated by a hypersurface (edge of chaos). On this surface the flow develops neither towards the laminar nor towards the turbulent state, but to an attractor within the surface that is called the edge state. During transition to turbulence, the flow may first reach the vicinity of the edge state before it eventually becomes turbulent. Edge states correspond to non-trivial time-dependent flows that have simpler dynamics than turbulent solutions. They can therefore serve as models for studyingthe physical mechanisms of transition. Edge states can, e.g., correspond to wave-like, time-periodic or chaotic solutions. They were studied in detail for a number of prototypical shear flows, but have hardly ever been considered for magnetohydrodynamic (MHD) flows although transition in these flows is also sub-critical. In MHD flows, the Hartmann number Ha appears as a second parameter for the magnetic field in addition to the Reynolds number as a dimensionless parameter for the velocity. In the project, the route to turbulence and the mechanisms of transition in MHD duct flows will be determined that are largely unknown. The specific configurations to be studied are flows in ducts with rectangular cross-section and transversal homogeneous magnetic field. These flows are three-dimensional and characterized by boundary layers on the walls parallel and perpendicular to the magnetic field that are calledShercliff and Hartmann layers, respectively. The flows will be investigated by highly resolved direct numerical simulations (DNS) and stability analysis. First, the boundary between laminar and turbulent states will be determined in the Re-Ha-A-parameter space, where A is theperturbation amplitude of the initial state. In the second step, edge states will be determined by DNS and their dynamics and specific properties will be analyzed. In the third step, the bifurcations of flow states upon changing of Re and Ha will be investigated that give rise to edge states. By that one can also obtain a better understanding why transition is determined by a parameter that is based on the thickness of the Hartmann layers although turbulence appears first in the Shercliff layers. Our results are relevant for electromagnetic flow control in metallurgical and heat transfer applications with conducting liquids.