Stability and energy budget of pressure-driven collapsible channel flows

Although self-excited oscillations in collapsible channel flows have been extensively studied, our understanding of their origins and mechanisms is still far from complete. In the present paper, we focus on the stability and energy budget of collapsible channel flows using a fluid–beam model with the pressure-driven (inlet pressure specified) condition, and highlight its differences to the flow-driven (i.e. inlet flow specified) system. The numerical finite element scheme used is a spine-based arbitrary Lagrangian–Eulerian method, which is shown to satisfy the geometric conservation law exactly. We find that the stability structure for the pressure-driven system is not a cascade as in the flow-driven case, and the mode-2 instability is no longer the primary onset of the self-excited oscillations. Instead, mode-1 instability becomes the dominating unstable mode. The mode-2 neutral curve is found to be completely enclosed by the mode-1 neutral curve in the pressure drop and wall stiffness space; hence no purely mode-2 unstable solutions exist in the parameter space investigated. By analysing the energy budgets at the neutrally stable points, we can confirm that in the high-wall-tension region (on the upper branch of the mode-1 neutral curve), the stability mechanism is the same as proposed by Jensen and Heil. Namely, self-excited oscillations can grow by extracting kinetic energy from the mean flow, with exactly two-thirds of the net kinetic energy flux dissipated by the oscillations and the remainder balanced by increased dissipation in the mean flow. However, this mechanism cannot explain the energy budget for solutions along the lower branch of the mode-1 neutral curve where greater wall deformation occurs. Nor can it explain the energy budget for the mode-2 neutral oscillations, where the unsteady pressure drop is strongly influenced by the severely collapsed wall, with stronger Bernoulli effects and flow separations. It is clear that more work is required to understand the physical mechanisms operating in different regions of the parameter space, and for different boundary conditions

An Arnoldi-frontal approach for the stability analysis of flows in a collapsible channel

In this paper, we present a new approach based on a combination of the Arnoldi and frontal methods for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigensolution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system have been obtained which are beyond the limits of previously used methods

Fast and Adaptive Sparse Precision Matrix Estimation in High Dimensions

This paper proposes a new method for estimating sparse precision matrices in the high dimensional setting. It has been popular to study fast computation and adaptive procedures for this problem. We propose a novel approach, called Sparse Column-wise Inverse Operator, to address these two issues. We analyze an adaptive procedure based on cross validation, and establish its convergence rate under the Frobenius norm. The convergence rates under other matrix norms are also established. This method also enjoys the advantage of fast computation for large-scale problems, via a coordinate descent algorithm. Numerical merits are illustrated using both simulated and real datasets. In particular, it performs favorably on an HIV brain tissue dataset and an ADHD resting-state fMRI dataset.Comment: Maintext: 24 pages. Supplement: 13 pages. R package scio implementing the proposed method is available on CRAN at https://cran.r-project.org/package=scio . Published in J of Multivariate Analysis at http://www.sciencedirect.com/science/article/pii/S0047259X1400260