Stability, nolinear regimes and transport properties of viscous flows subject to spatially inhomogeneous forcing: theory and applications.

近年来，流体力学对具有空间规律性和周期、准周期或混沌时间动力学特征的多涡型引起了广泛的关注。由于原始均匀流动的不稳定性或直接外部影响的结果，这种模式在从宇宙学和大尺度大气现象到微流体中的涡旋流动的各种设置中都会遇到；在工业中，它们在冶金和化学技术中有应用。在实验中，这种模式已经在液态金属和其他导电介质中通过空间中周期性的电流的作用重现。在理论分析中，一个典型的例子是开创性的Kolmogorov流，由空间周期性力激发，作为推断不稳定机制和理解湍流阶段级联能量传递的模型。正如进一步的研究（特别是参与该项目的研究人员）所证明的那样，将Kolmogorov设置推广到具有平均漂移的流动情况，并将一维强迫扩展到空间中双周期的固定力，从而导致了动态的新一类流动。这些流动处于层流和湍流之间的某种中间位置，并具有不寻常的性质：拉格朗日观测值的分形功率谱和被动掺合料输运的异常。项目参与者研究的另一种形成多涡准二维模式的机制是局部热源或表面活性物质附近的Marangoni对流。在该项目中，我们的目标是进一步研究这些现象，除了传统的流体动力学特性外，还将重点放在各种静止和随时间变化的涡流流型的光谱和输运特性上。我们将研究这些特性对流型结构、涡的相对强度和平均漂移、装置的几何形状和对称破坏效应的依赖。其他因素：由于流体旋转或由于电磁力对导电液体的作用而在流动中产生的三维性也将被考虑在内。我们期望问题设置中的各种修改最终会破坏固定的流动模式，首先导致不同尺度上规则漩涡的形成，导致它们的非线性相互作用，进一步导致流动中时间依赖性的开始，最后导致空间无序性的增长和湍流的开始。通过这种方式，我们的研究将加深关于在涡旋流中产生湍流的机制的理论知识，并在实践方面有助于在应用中更有效地利用此类流。

Recently, fluid mechanics has witnessed a raise of interest to multi-vortex patterns, characterized by spatial regularity and featuring periodic, quasiperiodic or chaotic temporal dynamics. Created either by the instability of primary uniform flows or as a result of direct external influence, such patterns are encountered in a variety of setups from the cosmological and large-scale atmospheric phenomena to vortical flows in microfluidics; in the industry they have applications e.g., in metallurgy and chemical technologies. Experimentally, such patterns have been reproduced in liquid metals and other conducting media by the action of electric currents, periodic in space. In theoretical analysis, a canonical example is the seminal Kolmogorov flow, excited by the spatially periodic force and serving as a model for inferring the mechanisms of instability and for understanding the cascade energy transfer at the turbulent stage. As demonstrated by further studies (in particular, by researchers, participating in this Project), generalizations of the Kolmogorov setup to the case of flows with mean drift, and extension from one-dimensional forcing to stationary forces that are doubly periodic in space, lead to a dynamically new class of flows. These flows occupy a certain intermediate position between laminar and turbulent ones and feature unusual properties: fractal power spectra of Lagrangian observables and anomalies in the transport of passive admixtures. Another mechanism for the formation of multi-vortex quasi-two-dimensional patterns, studied by the participants of the project, is the Marangoni convection near the localized heat source or the surface-active substance. Within the Project, we aim at further research of these phenomena, focusing, along with conventional hydrodynamical characteristics, at spectral and transport properties of various stationary and time-dependent flow patterns with vortices. We will investigate the dependence of these properties on the configuration of the flow pattern, on the relative intensity of the vortices and the mean drift, on the geometry of the setup and on the symmetry-breaking effects. Additional factors: onset of three-dimensionality in the flow as a result of the fluid rotation or due to the action of electromagnetic forces upon electroconductive liquids, will be taken into account as well. We expect that various modifications in the problem setup should eventually destabilize the stationary flow patterns, lead, first, to formation of regular eddies on different scales, to their nonlinear interaction, further, to the onset of time-dependence in the flow and, finally, to the growth of spatial disorder and the onset of turbulence. In this way, our research will deepen the theoretical knowledge about the mechanisms that generate turbulence in vortical streams, and, on the practical side, contribute to the more efficient usage of such flows in applications.