The main aim of this paper is to relate instability modes with modes obtained from proper orthogonal decomposition (POD) in the study of global spatio-temporal nonlinear instabilities for flow past a cylinder. This is a new development in studying nonlinear instabilities rather than spatial and/or temporal linearized analysis. We highlight the importance of multi-modal interactions among instability modes using dynamical system and bifurcation theory approaches. These have been made possible because of accurate numerical simulations. In validating computations with unexplained past experimental results, we noted that (i) the primary instability depends upon background disturbances and (ii) the equilibrium amplitude obtained after the nonlinear saturation of primary growth of disturbances does not exhibit parabolic variation with Reynolds number, as predicted by the classical Stuart–Landau equation. These are due to the receptivity of the flow to background disturbances for post-critical Reynolds numbers (Re) and multi-modal interactions, those produce variation in equilibrium amplitude for the disturbances that can be identified as multiple Hopf bifurcations. Here, we concentrate on Re = 60, which is close to the observed second bifurcation. It is also shown that the classical Stuart–Landau equation is not adequate, as it does not incorporate multi-modal interactions. To circumvent this, we have used the eigenfunction expansion approach due to Eckhaus and the resultant differential equations for the complex amplitudes of disturbance field have been called here the Landau–Stuart–Eckhaus (LSE) equations. This approach has not been attempted before and here it is made possible by POD of time-accurate numerical simulations. Here, various modes have been classified either as a regular mode or as anomalous modes of the first or the second kind. Here, the word anomalous connotes non-compliance with the Stuart–Landau equation, although the modes originate from the solution of the Navier–Stokes equation. One of the consequences of multi-modal interactions in the LSE equations is that the amplitudes of the instability modes are governed by stiff differential equations. This is not present in the traditional Stuart–Landau equation, as it retains only the nonlinear self-interaction. The stiffness problem of the LSE equations has been resolved using the compound matrix method.
A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorithm with two benefits. First, the number of overlap points is reduced to a single common boundary point between any two neighboring sub-domains, thereby saving the number of points used, with resultant speed-up. Second, with a proper design, errors arising due to sub-domain boundary closure schemes are reduced to a user designed error tolerance, bringing the new parallel scheme on par with sequential computing. Error reduction is achieved by using global spectral analysis, introduced in “Analysis of central and upwind compact schemes,” J. Comput. Phys. 192, 2, (2003) 677--694, which analyzes any discrete computing method in the full domain integrally. The design of the parallel compact scheme is explained, followed by a demonstration of the accuracy of the method by solving benchmark flows: (1) periodic two-dimensional Taylor-Green vortex problem; (2) flow inside two-dimensional square lid-driven cavity (LDC) at high Reynolds number; and (3) flow inside a non-periodic three-dimensional cubic LDC with the staggered grid arrangement.
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