Session ZC25: Flow Instability: Nonlinear Dynamics

Intermittency route to chaos in a forced globally unstable jet

We explore the universal transition to chaos in a prototypical hydrodynamic oscillator, namely a globally unstable low-density jet subjected to time-periodic acoustic forcing. We find that as the forcing strengthens at an off-resonant frequency, the jet exhibits a sequence of nonlinear states: a period-1 limit cycle $ ightarrow$ $mathbb{T}^2$ quasiperiodicity $ ightarrow$ intermittency $ ightarrow$ deterministic chaos. We confirm the existence of chaos through the 0--1 test, the correlation dimension, and the horizontal visibility graph. We then show that the intermittency obeys type-II Pomeau--Manneville dynamics by analyzing the first return map, the recurrence plot, and the scaling laws of the quasiperiodic epochs between successive bursts of chaos. By providing experimental evidence of the type-II intermittency route to chaos in a globally unstable jet, this study reinforces the notion that strange attractors emerge via universal mechanisms in open self-excited flows. This discovery paves the way for the development of instability control strategies based on chaos theory.   

Nonlinear energy transfer in vortex-pairing of initially-laminar jets

In this study, we investigate the vortex-pairing process and its associated nonlinear energy transfer in the shear layer of an initially-laminar jet. Large-eddy simulations (LES) of a turbulent and initially-laminar jet are performed. Compared to the turbulent jet, the initially-laminar jet develops later but at a faster rate, resulting in a shorter potential core length and a hump in the RMS velocity along the centerline. The latter is caused due to the vortex-pairing in the shear layer. The vortex-pairing process involves two distinct occurrences: ($i$) two vortices of $St =1.76 $ pair to form an $St=0.88$ vortex, and ($ii$) two $St = 0.88$ vortices pair to form an $St=0.44$ vortex. Using local stability theory, we identify the fundamental as the most unstable frequency, which is $St=1.76$. Next, we evaluate the energy transfer between different frequencies, based on the nonlinear energy transfer term in the spectral turbulent kinetic energy equation. For this purpose, we employ bispectral mode decomposition (BMD), a technique that measures the intensity of triadic interactions. Our findings show that the energy is transferred from the fundamental to its subharmonic, resulting in the growth of the subharmonic. Additionally, energy transfer from the first subharmonic to its second subharmonic leads to the growth of the second subharmonic.  These energy transfers are primarily driven by two dominant triads, ($St_1,St_2,St_3$) = (1.76,-0.88,0.88), and (0.88,-0.44,0.44). Our results are consistent with previous work by citet{monkewitz1988subharmonic}, which demonstrated that the transfer was due to a resonance mechanism between the fundamental and subharmonic. The energy transfer during vortex-pairing exhibits characteristics of an inverse energy cascade.   

Nonlinear optimal perturbation analysis of quasi-periodic flow around an airfoil

The nonlinear optimal perturbation (NLOP) analysis is a stability analysis method to extract the most growing disturbance at given evaluation time around the base flow. While the previous studies focused on the growth of small initial perturbation in theflow field, we consider the case with large magnitude of initial perturbation in the compressible external flow. For this purpose, the NLOP analysis method for large initial disturbance norm was newly formulated. The method was applied to the problem of the trailing edge noise phenomenon around the NACA 0012 airfoil. This trailing edge noise is known to be a quasi-periodic phenomenon in the subsonic andmoderate Reynolds number regime. The obtained nonlinear optimal perturbation showed a series of disturbance vortices on the suction side of airfoil like the two-dimensional Tollmien-Schlichting wave. This disturbance showed a similarity to that of NLOP for the circularcylinder as well as the formation of laminar separation bubble. The dependence of the NLOP on the evaluation time and the magnitude of the initial disturbance was investigated in relation to the induced fluid instability. The transient disturbance growth was also analyzed.   

Optimally time-dependent mode analysis of vortex gust-airfoil wake interactions

Understanding transient dynamics in flows with time-varying base states is critical for evaluating the performance of wings in gusty environments. In this study, optimally time-dependent (OTD) modes, a set of orthonormal modes that traces the most amplified directions of the dynamics with respect to time-varying base flow, are considered to reveal the primary transient flow features. Instead of identifying the asymptotic behaviors, the non-linear trajectory of highly unsteady flows is tracked in a real-time manner.

We apply the OTD mode analysis to gust vortex-airfoil wake interactions, which exhibit strong transient amplification as the gust vortex impacts the airfoil. The OTD modes capture the dominant amplification dynamics stemming from the shear layer and the wake. For both positive and negative vortex-airfoil interactions, positive instantaneous growth rates are captured from the reduced-order linear operators in the OTD subspace.  Moreover, we uncover that the most amplified direction associated with the largest growth rate is closely related to the largest magnitude of lift force variation experienced by the airfoil.   

Unfolding the weakly-nonlinear dynamics of polyhedral flames

Under certain conditions, conical lean premixed flames spontaneously break their continuous azimuthal symmetry to form corrugated polyhedral flames with discrete symmetry. In particular, such polyhedral shapes are commonly observed in hydrogen flame experiments. In our earlier works, we have shown that this phenomenon is driven by the intrinsic flame dynamics via a global linear instability with zero frequency that occurs at sufficiently low Lewis number and/or sufficiently high Damköhler number. In this work, we perform a center manifold reduction for this circle-pitchfork (CP) type bifurcation in order to elucidate the weakly-nonlinear dynamics of the transition. To start, we study the case of a single bifurcating polyhedral mode, where the universal unfolding reveals the global spatiotemporal nonlinear dynamics in the vicinity of the critical point. We then move on to a bimodal case where two modes of different azimuthal symmetries simultaneously bifurcate from a codimension-2 double-CP point. In this case, the analysis reveals complex global dynamics that include mixed modes, which may be linked to experimental observations of spontaneously rotating polyhedral flame structures.   

Computational Examination of Stress Relaxation and Shear Banding in Sheared Amorphous Materials

Shear banding occurs in a variety of amorphous materials when yielding under shear deformation, from food and cosmetic products to cements and soils, where only parts of the material flow while others remain "stuck". While this phenomenon is broadly observed, its microscopic origins and its link to yielding transitions are not well understood. Using data from large-scale 3D computer simulations [1,2], which model non-Brownian jammed suspensions of soft, spherical particles, we examine the influence of the deformation rate and sample age on the relaxation of stresses and the onset of banded flow. Load curves are analyzed to characterize the severity of the stress drops, while profiles of the particle velocities are used to examine the development of shear bands. Theoretical models [3] have recently demonstrated the dependence of the severity of the stress drop and the development of shear bands in these systems on the history of the material and the shear rate. Through our computational analysis, we demonstrate an increase in the severity of the stress drop and the degree of banding as shear rate decreases and sample age increases, which agree with theoretical predictions. At high rates, these materials begin to yield and flow in a uniform, ductile fashion, while at low rates we see a sudden, severe drop in shear stress, or brittle yielding, which leads to the development of strong, and persistent, shear banding.   

Nonlinear evolution of disturbances entrained in a circular pipe

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the long-wavelength disturbances which induce streamwise elongated streaks. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances and their amplitude is assumed to be intense enough for nonlinear interactions to occur. The formation and evolution of the streaks are described by the nonlinear unsteady boundary-region equations written here in cylindrical coordinates for the first time. Supplemented by appropriate initial and boundary conditions, this initial-boundary-value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilizing effect of nonlinearity on the intense algebraic growth of the streaky structures. For a high free-stream turbulence level, all the disturbances attenuate sufficiently downstream with the exception of a pulsating mode. A parametric study is carried out to evince the effect of Reynolds number, streamwise and azimuthal wavelength, and radial characteristic scale on the nonlinear evolution. Qualitative agreement between our numerical results and the limited experimental data is obtained.   

Transverse instability of autocatalytic fronts in radial injection

A cubic autocatalytic reaction-diffusion front is a localized reaction zone traveling at a constant speed with a fixed shape [1, 2]. The autocatalytic front properties are affected by the flow rate at early time when the autocatalytic species X is radially injected into a pool of the reactant Y [3]. In the reverse case where the reactant Y is injected at a constant flow rate into X, “frozen fronts” i.e. static fronts maintained at a fixed radius can be obtained since the injection of fresh reactant counterbalances the inward invasion by the autocatalytic species. Analytical solution for concentrations in this stationary regime is obtained, and the theoretical predictions for the front position agree well with experiments [4]. We show numerically that, when the invading reactant Y has a sufficiently larger diffusivity than the displaced autocatalytic species X, a transverse diffusive instability can induce modulated structures with a given number of petals in the θ-direction around the stationary front location. Space-time maps of the dynamics reveal that positions of the crests and troughs of the wavy front vary over time. Simulations also show that an increase in the injection speed and diffusivity coefficient of the invading fluid decreases the transverse modulation’s wavelength.

 

[1] S. K. Scott and K. Showalter, J. Phys. Chem 96, 8702 (1992).

[2] A. De Wit, Phys. Rev. Lett. 87, 054502 (2001).

[3] A. Comolli et al., Phys. Chem. Chem. Phys. 25, 10604 (2023).

[4] L. Negrojević, A. Comolli, F. Brau, A. De Wit (preprint).   

Three-dimensional simulations of gravity currents in cold, fresh water

Counter-intuitively, some fluids attain a maximum density at a temperature above their freezing point. Water is one such fluid. As a result, the density of cold (<10°C), freshwater systems have an effectively quadratic dependence on temperature. Researchers have observed that this nonlinearity impacts the profile, speed, and shear instabilities of two-dimensional gravity currents in cold, fresh water. We extend this work to three-dimensional systems with no-slip boundary conditions. This allows us to study the lobe-cleft instability: an inherently three-dimensional instability that produces dynamic patterns of folds and lumps along the current front. In this talk we will discuss how the lobe-cleft instability is modulated by the nonlinear equation of state. We will also discuss how the lobe-cleft instability three-dimensionalizes the billows produced by the shear instability, and how this, too, is affected by the nonlinear dependence on temperature. Our major result is that effects on the lobe-cleft instability do not manifest until secondary instabilities develop. Finally, we will briefly discuss how our results may extend to the hydrology of ice-covered lakes.   

Multiple steady states and symmetry-breaking in stratified anabatic flows in idealized valleys

Due to evening cooling of the atmosphere, sloping terrain experiences downslope, or katabatic, winds during the nighttime, which lead to the formation of stably stratified cold pools in the base of valleys. Subsequently, morning heating causes upslope, or anabatic, flows, leading to the destruction of the stratified layer. However, the specific dynamics of these stratified anabatic flows in valleys is not well understood. In this study, we characterize the full structure of steady laminar anabatic flows in a stably stratified V-shaped valley using a dynamical systems approach. Our approach is based on the discovery of a quiescent conduction state from which a unique asymmetric nested-pitchfork bifurcation emerges. With sufficient surface heating, the conduction state bifurcates into two possible states, including symmetric and asymmetric steady state profiles. The asymmetric state manifests as a mirror image pair with clockwise and counterclockwise central circulations, while the symmetric state gives rise to upslope and downslope convection patterns, which do not satisfy the same mirror image reflection. Linear modal analysis and numerical simulations show that these two symmetric states are linearly unstable and will transition to the asymmetric state under the slightest perturbation.