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的研究越来越多。这种模式是由初级均匀流动的不稳定性或直接的外部影响造成的，在从宇宙学和大尺度大气现象到微流体中的涡流的各种设置中都会遇到这种模式；在工业中，它们有应用，例如在冶金和化学技术中。在实验上，这种模式已经在液态金属和其他导电介质中通过电流的作用在空间中周期性地重现。在理论分析中，一个典型的例子是开创性的Kolmogorov流，它由空间周期性的力激发，并作为一个模型来推断不稳定的机制和理解湍流阶段的叶栅能量传递。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集合推广到具有平均漂移的流动的情况，以及从一维强迫延伸到空间中双周期的定常力，导致了一种动态的新的流动类型。这些流动介于层流和湍流之间，具有不同寻常的性质：拉格朗日可观测量的分形功率谱和被动混合物输运中的异常现象。项目参与者研究的另一种形成多涡旋准二维模式的机制是局域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，除了常规的流体动力学特性外，还将重点放在各种带有涡流的定常和随时间变化的流型的光谱和输运特性上。我们将研究这些性质与流型的构型、涡旋和平均漂移的相对强度、装置的几何形状以及对称破缺效应之间的关系。其他因素：由于流体旋转或由于电磁力对导电液体的作用，流动中的三维开始也将被考虑在内。我们预计，问题设置中的各种修改将最终破坏定常流型的稳定性，首先，导致不同尺度上规则涡的形成，它们的非线性相互作用，进一步，导致流动中的时间依赖性的开始，最后，导致空间无序的发展和湍流的开始。这样，我们的研究将深化关于涡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这种流动。

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.