Topology of corner vortices in the lid-driven cavity flow: 2D vis a vis 3D
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O R I G I NA L
Sougata Biswas · Jiten C. Kalita
Topology of corner vortices in the lid-driven cavity flow: 2D vis a vis 3D
Received: 12 November 2019 / Accepted: 23 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract All the previous studies on the cavity flow are confined to either the study of its 2D or its 3D configuration in isolation. In this study, we endeavour to gain some physical insight into the corner vortices from the perspective of the flow topology in the 2D vis a vis 3D driven cavity by employing some recent developments in the field of topological fluid dynamics. The computed flow is post-processed to identify critical points in the flow field leading to the prediction of separation, reattachment and vortical structures in the flow. The limit cycles in the plane of symmetry of the 3D flow representing the vortices are found to be stable ones. The Poincaré–Bendixson formula is used to validate the computed flow, i.e., the possible number of critical points in the 2D cavity identified by us from the computation. The topology of the corner vortices in actual 3D flow and its 2D idealization has also been compared in detail. Keywords Driven cavity · Corner vortices · Topological fluid dynamics · Limit cycles · Poincaré–Bendixson formula
1 Introduction In the last few decades, the flow in the lid-driven cavity [4,8,10,25] is probably the most widely used problem amongst the computational fluid dynamics community as a benchmark problem for establishing newly developed numerical schemes [10,14–17,26]. Despite being set in simplest of geometric settings, the flow in the driven cavity exhibits a plethora of fluid flow characteristics such as vortex dynamics, hydrodynamics instability and bifurcations. All these flow characteristics are mainly manifested through the presence of multiple counter-rotating corner eddies at the corners of the cavity depending on the Reynolds number (Re). It is worth mentioning that in almost all the studies available in the literature on this topic, 2D flows as an idealization of a 3D ones or 3D flows having symmetry in one direction have been considered. The present study is not different from them where one expects the flow to be exactly the same found in the plane of symmetry of a 3D lid-driven flow in a rectangular cavity with infinite span-wise length. Figure 1a shows a simple 2D lid-driven square cavity defined on 0 ≤ x, y ≤ 1, the configuration of which may be considered as the idealization of the flow in the plane of symmetry in a 3D lid-driven rectangular cavity with infinite spanwise length. Such a flow can be visualized in Fig. 1b, where one can see the flow in the plane of symmetry of a 3D rectangular cavity with span-wise aspect ratio 2. Note that all the previous studies on the cavity flow are confined to either the study of its 2D or its 3D configuration in isolation. To the best of our knowledge, no study of the cavity flow in the context of 2D vis a vis 3D is available in the literature. The objective of the current study is to prov
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