Universal instability modes in internal and external flows

Abstract

Two prototypical external and internal flows have been studied which display linear temporal instability followed by nonlinear saturation taking the flows to a new equilibrium state. Direct simulation results are obtained using a specific formulation and numerical methods with very high accuracy. These results are analyzed via proper orthogonal decomposition (POD), which reveal similar modes for flow past a circular cylinder and flow inside a lid-driven cavity, indicating universality of such modes. Unlike many other efforts on reduced order modeling via POD, here the emphasis has been on understanding the physical aspect of the flow instability which requires very high accuracy of the simulation. Then, the obtained POD modes are related to the instability modes (in the classical sense of defining the latter) and new generic types of instability modes are identified in the studied external and internal flows. These new modes have been reported for flow past a circular cylinder [Sengupta TK, Singh N, Suman VK. Dynamical system approach to instability of flow past a circular cylinder. J Fluid Mech 2010;656:82–115] which help one in understanding the instability sequence and the relative importance of these modes in the flow evolution starting from an impulsive start. Present comparative study, furthermore, reveals universality of such instability modes by showing their presence for the flow inside a lid-driven cavity as well. Despite seeming dissimilarities between these two flows, similarities between the instability portrait of these two flows suggest universality of such modes. From the equilibrium amplitude of vorticity time-series, we establish the presence of multiple modes and multiple bifurcation sequences for these flows in parameter space. Existing theory due to Landau and Stuart that considers only a single dominant mode and its self-interaction does not explain all these features. We invoke a multi-modal interaction model in the cited reference above, termed as Landau–Stuart–Eckhaus (LSE) equation in recognition of Eckhaus’ work in modifying the classical Stuart–Landau equation. We also show that the new instability modes do not follow either the classical Stuart–Landau or the newly proposed LSE model equations and for this reason we call these as anomalous modes. Two specific classes of anomalous modes are identified and classified in the present work. Empirical expressions for the evolution of these anomalous modes are presented and their unambiguous role during instabilities is discussed.