Session P08: Flow Instability: Theory and Nonlinear Dynamics

Flow past a linearly-sprung, freely-rotatable circular cylinder with an eccentric center of mass

Previous work (Tumkur et al., J. Fluid Mech. 828, 196-235, 2017; Blanchard et al. Phys. Rev. Fluids, 4, 054401, 2019; Blanchard & Pearlstein, Phys. Rev. Fluids, 5, 023902, 2020) for crossflow past a linearly-sprung (nonrotatable) circular cylinder with an attached rotatable mass allows multiple unsteady, long-time solutions at Reynolds numbers (based on cylinder diameter) below 50, and that locking or releasing the attached mass can be used as an onn/off switch for vortex-induced vibration (VIV).  Here, we consider a similar case, but with a freely-rotatable cylinder whose center of mass lies off-axis, with no separate attached mass.  The cylinder is allowed to vibrate in the cross-stream direction, and to rotate about its geometric axis driven by torque due to the flow, and damped by shaft friction (with damping proportional to the angular velocity).  Computations over a range of dimensionless parameters characterizing the spring constant, the damping associated with rotation about the shaft, and the mass distrbution in the cylinder show that the stability properties of the steady, symmetric motionless-cylinder solution (with no VIV) are reasonably similar to those found previously, but that the dynamical behavior of the solutions arising from the instability is quite different.   

Not Participating

We present a framework for computing nonlinear sensitivities using a model-driven low-rank approximation. To this end, we use the model's nonlinear perturbation equation (NLPE) to derive evolution equations for a low-rank orthonormal spatial basis, low-rank correlation matrix, and low-rank orthonormal parametric basis. While solving the full-rank system scales linearly with the number of perturbations, this reduced framework directly solves for the intrinsic low-dimensional manifold by leveraging correlations between perturbations on the fly. Therefore, the cost of solving the low-rank approximation scales linearly with the intrinsic rank, which allows for efficient computation of the NLPE for finite perturbations in a high-dimensional parametric space. Furthermore, in contrast to linear sensitivity equations, the NLPE remains stable for chaotic systems. We will present a case study for chaotic and non-chaotic flow with finite perturbations in the 2D compressible Navier-Stokes equations.   

Weak nonlinearity for strong nonnormality

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of transiently growing and harmonically forced nonnormal systems. This reconciles the non-modal nature of these growth mechanisms and the need for a center manifold to project the leading order dynamics. Under the hypothesis of strong nonnormality, we demonstrate that small operator perturbations suffice to let respectively the inverse resolvent and the inverse propagator become singular. The methodology is outlined for a generic nonlinear dynamical system and two application cases are chosen which highlight two common nonnormal mechanisms in hydrodynamic instability: a backward-facing step prone to streamwise convective nonnormality and a plane Poiseuille flow subjected to lift-up nonnormality.   

Spontaneous symmetry-breaking of geometrically confined Faraday waves

When a fluid bath is vibrated vertically, there exists a critical driving acceleration beyond which the flat hydrostatic surface becomes unstable and transitions to a field of subharmonic standing waves, the so-called Faraday waves. A further increase in forcing will see the system undergo a secondary instability in which the standing Faraday pattern spontaneously becomes chaotic. When the bath is large with respect to the characteristic wavelength, the waves form elongated patterns that appear, drift and disappear randomly on the free surface. We demonstrate that, by geometrically confining these waves, their chaotic motion can be harnessed to produce coherent directed motion. In particular, in annular geometries, these chaotic waves can spontaneously develop into fast-moving travelling waves in either clockwise or anti-clockwise directions. We rationalize the mechanism responsible for this new instability in terms of the streaming flows generated at the boundary layers on the vertical walls, which are critically enhanced by the presence of a meniscus. Potential applications of these out-of-equilibrium waves will be discussed.   

Estimating stability margins for transitional flows using quadratic constraints

Laminar to turbulent transition can be detrimental to many engineering systems.  Yet, predicting precisely when a flow will transition has been notoriously difficult.  In this work, we propose a method for estimating stability margins of nonlinear fluid systems.  The approach considers the stability of the linear dynamics subject to quadratic constraints that describe input-output properties of nonlinear terms.   We demonstrate the approach on the Burgers equation and a reduced-order model of plane Couette flow.  Our results indicate that the proposed method reduces conservatism in estimating regions of attraction and permissible perturbation amplitudes compared to related quadratic-constraint-based analysis techniques. Additionally, we demonstrate that the approach, while introducing some conservatism, has lesser computational complexity than commonly used stability analysis techniques, including sum-of-squares optimization and direct-adjoint looping methods.    

Folding points, flow instabilities and onset of vortex formation

A Lagrangian finite-time diagnostic to identify distinct points of fluid material folding is introduced and applied in the analysis of several shear flows. By considering the deformation of lines of fluid particles over a finite integration time interval, their curvature change is used to quantify the strength of the fluid folding and central locations of the wrinkling at the onset of vortex formation. Because the curvature-based analysis combines the stretching and rotation of the fluid elements, distinct and subtle structures in the flow field can be highlighted after shorter integration intervals compared to classic strain-based Lagrangian visualization techniques, such as the finite-time Lyapunov exponent.

The approach is tested to capture the vortex formation in a temporally developing jet flow, an unstable separated shear flow over a cambered airfoil, and in the onset of a wake instability behind a circular cylinder. The kinematic finite-time approach has practical benefits for experiments that have only limited data available, and could potentially be applied to snapshots of streak lines that are recorded over a convenient time window.   

Not Participating

This study addresses the interplay between hydrodynamic instabilities and a transmembrane flow, under the coupling by the osmotic pressure driven by a concentration boundary layer building up at the membrane. The configuration consists of a Taylor-Couette cell filled with solvent carrying a passive scalar. Whereas the concentric inner and outer cylinders are permeable to the solvent, they totally reject the scalar. As a radial flow of solvent is superimposed across the cell, a concentration boundary layer builds up at one of the cylinders. Besides, the rotation of the inner cylinder drives centrifugal instabilities stirring the concentration boundary layer.

By mean of the osmotic pressure, the concentration boundary layer promotes the instabilities. In return, the instabilities increase the mean radial flow across the cell. The impact of the different physical parameters on these two mechanisms is then addressed.

These results obtained by analytical stability analyses are in agreement with dedicated Direct Numerical Simulations. They quantitatively assess the roll of hydrodynamic instabilities to improve the performances of filtration devices.   

Active learning of nonlinear operators via neural nets for forecasting instabilities leading to extreme events in fluids

Through approximations of nonlinear operators via neural networks, we develop a framework for computing predictors of extreme events (i.e. instabilities), in infinite-dimensional systems with applications to fluids. Extreme phenomena or instabilities, such as pandemic spikes, electrical-grid failure, or rogue waves, have catastrophic consequences for society. Unfortunately, characterizing extreme events is difficult because of their rarity of occurrence, the infinite-dimensionality of the dynamics, and the stochastic perturbations that excite them. These challenges are problematic as standard training of machine learning models assumes both plentiful data and moderate dimensionality. Neither is the case for extreme events. To navigate these challenges, we combine a neural network architecture designed for approximating infinite-dimensional, nonlinear operators with novel training schemes that actively select data for characterizing extreme events. We apply and assess these methods to prototype systems for deep-water waves having the form of partial differential equations. In this case, the extreme events take the form of randomly triggered modulation instabilities. Finally, we conclude by discussing the generality of this approach for modeling other extreme phenomena.   

Sparsity-promoting methods for automated simplification of linearized operators in fluid mechanics

Linear energy amplification mechanisms can play in important role in even highly nonlinear, turbulent flows. In many cases, mechanisms leading to large energy amplification arise due to the properties and interactions of a limited set of terms within the linearized equations. This work develops and applies a method to identify the minimal set of such terms that preserve the energy amplification properties of the original equations. This is done through the use of sparsity-promoting techniques, formulating an optimization problem that penalizes both the difference between the amplification properties of the true and simplified dynamics, and the number of retained terms in the simplified equations. The simplified operators that are identified through this procedure typically have very similar pseudospectral properties as the original equations, but with only the most important active terms present. We demonstrate the use of this method on both incompressible and compressible parallel shear flows, and relate the results to known amplification mechanisms. We additionally discuss the potential applicability of this method for a wider class of fluid flows.   

Pseudospectral analysis of parallel shear flows using wavepackets

The linearized Navier-Stokes equations for shear flows are typically highly nonnormal, motivating the use of pseudospectral analysis to study linear energy amplification mechanisms. This analysis can be performed via the singular value decomposition of the associated resolvent operator, for specified temporal frequencies and spatial wavenumbers. Here, we investigate an alternative approach to perform this analysis, relying on an assumption that spatial structures associated with large linear energy amplification can be efficiently represented as a sum of wavepackets of a specified form. Thus, this methodology can be recast as an optimization problem for a small number of parameters, and can be assisted by first identifying simplified operators that retain the pseudospectral properties of the original system. We demonstrate that the method can accurately predict mode shape and amplification levels for a variety of mode types, including cases where the modes are attached to one or two boundaries, and/or influenced by multiple critical layers. We additionally show that this methodology can also be successfully applied to the analogous study of suboptimal modes. The method produces closed-form analytic expressions for spatial mode shapes, which can be convenient for ensuing analysis.   

Input-output analysis of stochastic base flow uncertainty in channel flows

We adopt an input-output approach to study the effect of base flow perturbations on the stability and receptivity properties of transitional and turbulent channel flows. Base flow perturbations are modeled as persistent white-in-time stochastic excitations that enter the linearized dynamics as multiplicative sources of uncertainty that can alter the mean-square properties of state. We provide verifiable conditions for mean-square stability and study the frequency response of the flow subject to additive and multiplicative sources of uncertainty. Our approach does not rely on costly stochastic simulations or adjoint-based sensitivity analyses. We use our framework to uncover the Reynolds number scaling of critically destabilizing variance levels of the base flow uncertainty and identify the length-scales that are predominantly affected by perturbations of various shapes and sizes.   

Floquet analyses of complex 3D flows

Stability analyses of two-dimensional flows have become quite common during the past two decades. With increasing computational power, the stability of three-dimensional flows is feasible as well. However, much less attention has been paid to the stability of time-periodic flows. Such flows may arise due to harmonic forcing or naturally via a Hopf bifurcation. Furthermore, as the control parameter (e.g., Reynolds number) varies, these time-periodic dynamics may themselves become linearly unstable. Investigating these stability properties falls into the framework of Floquet theory.

This talk illustrates how an existing time-stepping code can be adapted to compute unstable periodic orbits and study their stability. It relies on nekStab, our recent open-source toolbox for large-scale stability analyses in Nek5000. After describing the algorithms implemented in nekStab, its capabilities will be illustrated on three examples: the flow past a tandem of side-by-side cylinders, the harmonically forced axial jet, and a three-dimensional pulsated jet in a cross-boundary layer flow.