Session ZC38: Vortex Dynamics and Vortex Flows: Instability

Nonlinear Stability and Evolution of 3D Vortices in Rotating Stratified Flows

Coherent, long-lived vortices play an important role in mixing and transportation in many geophysical and astrophysical flows. To understand their dynamics, previous work explored the linear stability of 3D, axisymmetric, Gaussian vortices in rotating, stratified flow. Building on this, the finite-amplitude stability and nonlinear evolution of linearly unstable vortices is studied using an initial value code solving the Boussinesq equations. The vortices studied have exponential growth rates faster than 0.2x their turn-around time and are seeded with a range of small initial perturbations with different spatial symmetries and amplitudes. Although their linear instabilities have fast growth rates, in all but one case the initial finite-amplitude perturbation quickly plateaus and the vortex evolves to a final steady, non-axisymmetric vortex nearly indistinguishable from its initial form. Thus, despite their broken symmetry, for most practical purposes these vortices are effectively stable. Mathematically put, the vortices have large effective Landau constants that, despite their eigenmodes' large growth rates, damp their linear amplitudes quickly. The results open questions for further study and may be relevant to other long-lived vortex families (e.g., aircraft wake vortices). All vortices studied have the ratio of the Coriolis parameter to Brunt-Vaisala frequency of the far-field flow set to f/N=0.1 and exist in the Rossby–Burger number parameter space -0.5<Ro<0.5 and 0.07   

Three-Wave Resonance in Neutrally Stable Wake Vortices

We examine the weakly nonlinear dynamics of linearly-neutrally-stable columnar vortices, whose nonlinear instabilities could help mitigate aircraft wake hazards. Generally, neutral eigenmodes must be in resonance to interact nonlinearly, where the lowest-order resonance in the wake vortex flows involves triplets of eigenmodes, with quadratic nonlinearities. Adapting a multi-scale perturbation approach, the evolution of the resonant triads is found to follow the classical "three-wave equations", and secondary instabilities such as the elliptical instability are recovered by setting one of the modes to be dominant. When dissipation is absent, triadic resonance of non-singular modes obeys the conservation laws known as the Manely-Rowe relations, which bound the triad's amplitudes for all time unless its resonance is of an explosive kind. Using a spectral method for unbounded domains and non-degenerate perturbation theory, the resonant triads of non-singular inviscid modes of the Lamb-Oseen vortex are quickly located, but none are found to be explosive. Therefore, we examine triads in which the dominant mode is a singular critical-layer mode that is regularized by viscosity, and we exploit a pseudospectral code that has tunable parameters to accurately and efficiently compute the regularized critical-layer modes.   

The Formation of the Secondary Vortex Street Instability

It is widely known that the near wake of bluff bodies is subject to the Karman vortex street instability. It is much less known that a qualitatively similar instability also exists in the far wake, in which large-scale vortices form, whether a near-wake Karman exists more upstream, or not. The cause of this downstream “secondary” vortex street instability is a source of controversy in the literature.

In this work, we investigate the secondary street in the far wake of solid plates (i.e. with Karman vortex shedding) and highly porous plates (i.e. without Karman vortex shedding). We perform a combination of LES and high-speed planar PIV at turbulent Reynolds numbers. The results are analyzed using the modal decomposition techniques. Our results suggest that the secondary vortex street formation is a sequential process, in which smaller vortices give their place to larger ones, and so forth. The effects of plate homogeneity and porosity on the resulting flow field are also discussed.    

Alliance of transient growth and sub-criticality in the Lamb-Oseen vortex: an amplitude equation approach

We analytically derive an amplitude equation for the weakly nonlinear evolution of the linearly most amplified response of a non-normal dynamical system. The development generalizes

a recently proposed operator exponential perturbation approach, in that the base flow here arbitrarily depends on time. Applied to the two-dimensional Lamb-Oseen vortex, the amplitude equation successfully predicts the nonlinearities to weaken or reinforce the transient gain. In particular, the minimum amplitude of the linear optimal initial condition required for a sub-critical bifurcation to happen in this flow is found to decay with the Reynolds number, which is confirmed by direct numerical simulations. The simplicity of the amplitude equation and the link made with the sensitivity formula suggests a physical interpretation of nonlinear effects, in light of existing work on Landau damping and on shear instabilities. The amplitude equation also quantifies the respective contributions of the second harmonic and the spatial mean flow distortion in the nonlinear modification of the gain.   

Investigation of Triggering Vortex Instabilities with Inertial Particles

In our previous study (Lee & Marcus, 2023, J. Fluid Mech.), we developed the mapped Legendre spectral method for analyzing wake vortices in an unbounded domain and obtained the linear eigenmodes of strong swirling q-vortices. A combination of these eigenmodes serves as an initial perturbation of the vortex, some of which may result in vortex destabilization as it nonlinearly grows. We present an extension of our numerical method in an Eulerian-Lagrangian framework that involves interactions between small inertial particles and the vortex structure. The effective computation of momentum exchange from the particles to the vortex and vice versa is discussed. By varying the initial position and velocity distribution of the particles, we investigate the evolution of the vortex perturbation resulting from the particle interaction. In particular, we examine whether the existence of particles is capable of creating viscous critical layers, which can trigger transient growth of the vortex, possibly leading to instabilities.   

Evolution of Vortex Lines in Transitional Pipe Flow

Concentrated vortex lines spontaneously arise in turbulent flows, acting as organizing features of the flow. In order to study the interactions between vortices and background flows in a controlled environment, we have built an experimental setup to generate concentrated streamwise vortices in pipe flow at for Reynolds numbers below 2000. We image the resulting flows in 3D using a combination of fluorescently dyed vortex cores and randomly distributed tracer particles. This allows for simultaneous reconstruction of the velocity field and a high fidelity tracking of the vortex shape. These experiments demonstrate a sharp transition in the vortex line behavior at Re ~ 1000, well below the full turbulent transition. The goal of this project is to shed new light on how vortices interact with walls and background flows, quantifying these interactions in terms of energy, helicity, and enstrophy.