Session Z18: Flow Instability: Nonlinear Dynamics & Control

Triadic nonlinear interactions in harmonically forced jets

Bispectral mode decomposition (BMD) is tailored to the extraction of flow structures associated with triadic interactions from spatio-temporal flow data. We use BMD to study the differences between the nonlinear dynamics of turbulent round jets with and without harmonic forcing. To this end, large-eddy simulations (LES) of unforced and axisymmetrically forced jets at $Re = 50,000$ and $M_j = 0.4$ are performed and validated with a companion experiments. Additional simulations are conducted for forcing azimuthal wavenumbers of $m=\pm 1$, and $m=\pm 6$ . The emphasis of the analysis is on the harmonics created by the fundamental forcing mode and its contribution to the mean flow deformation. Azimuthal wavenumber triads are investigated using the cross-spectral variant of BMD. Wu & Huerre (2009) have hypothesized that the interaction of a helical conjugate pair with $m=\pm 1$ is the most efficient radiator of low-frequencies sound to jet angles between $45^\circ - 60^\circ$, This hypothesis is the second objective of this contribution.

[1] Wu, X. & Huerre, P. 2009 Low-frequency sound radiated by a nonlinearly modulated wavepacket of helical modes on a subsonic circular jet. Journal of Fluid Mechanics 637, 173–211.

    

Author not Attending

This work addresses the data-driven forecasting of extreme events in airfoil flow. For certain Reynolds numbers and flow configurations, airfoils are subject to sporadic high amplitude fluctuations in the aerodynamic forces. These extreme excursions may be seen as prototypical examples of the kind of unsteady and intermittent dynamics relevant to the flow around airfoils and wings in a variety of laboratory and real-world applications. We build on the work of Rudy & Sapsis (2022) who explored a variety of machine learning models to learn the mapping from pressure measurements along the airfoil surface to the drag coefficient exhibiting the extreme events. In this work we investigate the spatial dependence of the temporal correlation and mutual information between the surface pressure sensors and the aerodynamic forces, with a specific focus on rare, high amplitude excursions of the later. These extreme excursions are found to be associated with the instabilities of certain temporal frequencies. We employ these findings to develop an algorithm to optimally place pressure sensors to efficiently learn the extreme event dynamics from a sparse distribution of sensors.  We then further exploit the previously mentioned instability mechanism to define and compute an observable which tracks the growth of locally unstable frequency components. This observable allows us to efficiently and accurately forecast imminent extreme excursions in the drag coefficient from observations of as little as a single pressure sensor.

    

Delayed Hopf bifurcation of a ferrofluid interface subject to a time-dependent magnetic field

Previously, we demonstrated that the combination of a radial and azimuthal static magnetic field deforms a confined ferrofluid droplet in Hele-Shaw cell into a stably spinning "gear". Weakly nonlinear analysis was used to predict the evolution. With the azimuthal field fixed, the traveling wave solution bifurcates from the trivial solution, as the radial field strength is increased. We show that this is a Hopf bifurcation at the critical growth rate: a positive linear growth rate allows for a stably spinning "gear", while a negative one leads to a stationary state. A center manifold reduction is applied to show the geometrical equivalence between a two-harmonic-mode coupled ODE system and the Hopf bifurcation. Inspired by the well-known delay behavior of time-dependent Hopf bifurcations, we design a slowly-time-varying magnetic field such that the timing of the emergence of the spinning "gear" can be controlled. The time-varying parameters and initial perturbation are determined through an amplitude equation derived from a multiple-time-scale expansion. This amplitude equation also reveals hysteresis in the time-dependent field manipulation.   

Revisiting the Ginzburg-Landau Equation: Model Reduction and Data-Driven Analysis of the Bénard-von Kármán Instability

The Stuart-Landau and Ginzburg-Landau equations model the temporal and spatial evolution of a small perturbation to a nonlinear dynamical system close to a Hopf bifurcation point in a dimensionless parameter, such as the Reynolds number Re. These equations are particularly relevant for describing the dynamics of the viscous wake behind a 2D cylinder, including the transition from a steady flow to time-periodic vortex shedding as Re increases through the critical bifurcation value Re_c ≈ 47. In this work, we investigate the use of data-driven sparse nonlinear modeling techniques to learn nonlinear ordinary and partial differential equations based purely on data from high-fidelity numerical simulations. In particular, we explore various coarse graining options to reduce the dimensionality and also compare the learned equations with the classical Stuart-Landau and Ginzburg-Landau models.   

Scalar mixing optimisation in 3D pulsatile channel flow

The mixing of passive scalars in energetic flows is a complex and important process, and have many applications in industrial and environmental contexts. In our study, we aim to utilise forcings that can be physically achieved in realistic situations to control the mixing process of scalars in three-dimensional Poiseuille flow. We note that pulsatile channel flow has recently been shown to exhibit destabilising effects at certain frequencies, with the strongest destabilisation seen at Womersley number Wo ≃ 7 (see Pier & Schmid (2017), Kern et al. (2021)). Floquet analyses are used to identify the most unstable eigenvalues for flows at different Womersley numbers. The associated eigenfunctions could then be used as the initial conditions for DNS simulations. Mixing efficiency for a range of oscillation parameters is investigated, and the effect of different nonlinear pulsatile flow regimes (namely the ‘ballistic’ and ‘cruising’ regimes) on the scalar mixing process is discussed. Through the use of variational “direct-adjoint looping” methods, it is also possible to potentially identify “optimal” initial perturbations for the homogenisation of the scalar field for a given forcing mechanism.

    

Arbitrary order Taylor expansions of the base flow and its eigenproblem

First order sensitivities and adjoint analyses are widely used today to assess the linear stability of unstable flows. Second order sensitivities have recently helped to increase accuracy. In this work, we calculate high order Taylor expansions of the base flow and its eigenproblem around a scalar parameter for the incompressible Navier-Stokes equations.

Two scenarios are investigated: first, for the two-dimensional cylinder flow, we expand along the inverse of the Reynolds number. We predict the base flow and its leading eigenvalue by Taylor-expanding around Re=47, which is the onset of flapping. With subsequent Taylor expansions, we calculate the base flow accurately until Re=1000, thus establishing a new method to calculate unstable base flows. We discuss the topic of convergence radii by evaluating predicted base flows until Taylor order 40 and show that the method is computationally very efficient.

Second, a small control cylinder is inserted into the computational domain of the cylinder flow for stabilization. For relatively low Reynolds numbers, stabilizing areas are accurately predicted by modeling the small cylinder with a steady forcing and drawing sensitivity maps up to Taylor order 10. The results are validated by incorporating the small cylinder directly into the grid.   

Continuous-time balanced truncation for time-periodic flows using frequential Gramians

Reduced-order models for flows that exhibit time-periodic behavior (e.g., flows in turbomachinery and wake flows) are critical for several tasks, including active control. The existing continuous-time balanced truncation method can be computationally expensive due to slowly-decaying transients and to the need to perform forward and adjoint impulse responses to compute the periodic reachability and observability Gramians. Moreover, this method can only be applied to systems that are stable in the sense of Floquet. To address these issues, we use the frequency-domain representation of the Gramians (henceforth referred to as frequential Gramians). First, these frequential Gramians are well-defined for both stable and unstable dynamics. Second, they can be estimated numerically by solving algebraic systems of equations that lend themselves to space-time parallelism and that deliver the desired post-transient response without following physical transients. We demonstrate our approach on a periodically-forced axisymmetric jet at Reynolds number Re=1500. In this configuration the flow exhibits heavy vortex pairing about an underlying unstable periodic orbit. We use balanced truncation for controller and observer design to restabilize the flow.   

Linear sensitivity of a hypersonic boundary layer to steady wall blowing and heating

Flow control efficiency depends on the location of the actuators. Instead of performing a computational costly parametric analysis, we use and adjoint-based optimisation technique to find the optimal actuator for steady open-loop control achieved through base-flow modification. Exploiting the benefit of Algorithmic Differentiation to ease the computation of high-order state derivative operators, it relies on the sensitivity of the most predominant global modes predicted by the resolvent analysis. The method is applied on a Mach 4.5 boundary layer over an adiabatic flat plate for steady wall-blowing control and on an isothermal flat plate for steady wall-heating control. The linear gradient predicted for the second Mack mode is studied in detail. The resolvent optimal gain decreases when suction is applied upstream Fedorov's mode S/mode F synchronisation point leading to stabilisation and conversely when applied downstream. The largest suction gradient is in the region of the branch I of mode S neutral curve. For the isothermal case, strong heating at the leading edge and cooling in the unstable region of mode S stabilises the second Mack mode. Sensitivity of the streaks and the first Mack mode is also briefly discussed.   

Multifractality and scale-free network topology in a noise-perturbed laminar jet

We present experimental evidence of multifractality and scale-free network topology in a noise-perturbed laminar jet operated in the unconditionally stable regime, prior to the critical point of a supercritical Hopf bifurcation and prior to the saddle-node point of a subcritical Hopf bifurcation. For both types of bifurcation, we find that (i) the degree of multifractality peaks at intermediate noise intensities, (ii) the conditions for maximal multifractality give rise to a complex network whose node degree distribution obeys a power-law scaling with an exponent of $2 < \gamma < 3$, indicating a scale-free network topology, and (iii) the Hurst exponent and the global clustering coefficient perform to different levels of effectiveness as early warning indicators of global self-excited instability. In characterizing the noise-induced dynamics of a canonical shear flow, we demonstrate that the multifractal and scale-free network dynamics often seen in turbulent flows can also be seen in a laminar flow under specific forcing conditions.   

Dynamics of Elliptical Vortices with Continuous Profiles

We examine the dynamics of elliptical vortices in 2D ideal fluid using an adaptively refined and remeshed vortex method. Four cases are considered: the compact MMZ and POLY vortices, and noncompact Gaussian and smooth Kirchhoff vortices (SK). The vortices have the same maximum vorticity and 2:1 initial aspect ratio, but unlike the top-hat Kirchhoff vortex, they have continuous profiles with different regularity. In all cases the co-rotating phase portrait has two hyperbolic points. At early time two filaments emerge and form a halo around the core as vorticity is advected along the unstable manifold of each hyperbolic point. The Gaussian vortex rapidly axisymmetrizes, but later on the core begins to oscillate and two small lobes emerge adjacent to the core; this is attributed to a resonance. For the MMZ, POLY, and SK vortices, the core maintains its ellipticity for longer time and the filaments entrain fluid into two large lobes forming a non-axisymmetric tripole state; afterwards the lobes repeatedly detrain fluid into the halo; this is attributed to a heteroclinic tangle. While prior work suggested that elliptical vortices evolve to either an axisymmetric state or a non-axisymmetric tripole state, our results suggest that such vortices may oscillate between these states.