Session Q11: Vortex Dynamics: Instability

Simplified model for helical vortex instabilities with applications to asymmetric rotor wakes

Helical vortices, such as those generated in the wake of a rotor, are subject to various instabilities including displacement instabilities, which occur when the vortex core is shifted from its baseline position. After being perturbed, the vortices deform and begin to break down. The zero-wavenumber displacement instability mode can be triggered by introducing an asymmetry to the rotor producing the vortices. The vortex dynamics in this case are highly complex, so a simplified model based on an infinitely-repeating strip of point vortices is developed to reproduce the nonlinear instability evolution. The model is validated against a more sophisticated filament model and water channel experiments, showing remarkable agreement for a range of parameters relevant to industrial rotors. It is then used to investigate the effectiveness of different types of rotor asymmetries at accelerating vortex breakdown. Even small initial displacements around 5% of the vortex spacing substantially disturb the vortices, and the direction of the perturbation plays an important role in the speed of the instability development. These findings can then be used to design wind turbine rotors that minimize the detrimental effects of their wakes on downstream turbines within a wind farm.   

Author not Attending

In this work, we present a 3D linear stability analysis of a radially stratified Rankine vortex of radius, r=a. The flow is assumed to be inviscid and the density distribution axisymmetric with a single density jump at radial location r=r_j. In an earlier study [1], it was shown that a light-cored vortex, i.e. ρ₁ < ρ₂, may become unstable while heavy-cored vortex, i.e. ρ₁ > ρ₂, may be stabilized. Instabilities were attributed to a linear wave-interaction mechanism between the density mode at r_j and a Kelvin mode at r=a. In this study, we extend the 2D stability results to include three-dimensional perturbations. For step density jumps at arbitrary radial locations outside the vortex core, we derive the complete dispersion relation analytically. The special asymptotic limit of r_j→a and ρ₁ = ρ₂ recovers the Kelvin's dispersion relation for a homogenous Rankine vortex. The dispersion relation also captures one family of continous spectrum modes of the Rankine vortex recently discovered [2]. Like the 2D case, we again find that a light-cored vortex can be unstable and a heavy-cored vortex stable.

References:

1. Dixit & Govindarajan, J. Fluid Mech., 679, 2011, 582-615.

2. Roy & Subramanian, J. Fluid Mech., 741 , 2014, 404-460   

Bursting on non-rectilinear vortex loops with initial core-size perturbations

Vortex tubes with initial core-size variations can develop twist waves and their collision leads to a sudden expansion of the vortex core, known as vortex bursting. In our earlier work, we analyzed vortex bursting on rectilinear vortex tubes using direct numerical simulations at circulation-based Reynolds number of 5000. Here we extend that work by analyzing numerical simulations of vortex bursting on non-rectilinear vortex loops, with specific focus on the effect of the core-line geometry on the bursting mechanics and vice-versa.   

Do rigorous bounds restrict the growth of vortex knots?

In 2018 the growth of the enstrophy for trefoil vortex knots provided weak empirical evidence for the formation of a finite-time energy dissipation, known as a dissipation anomaly JFM 839, R2, 2018. This was achieved using an algebraic, not Gaussian, profile of the vorticity about the centerlines of the vortex knot and allowing the size of computational domain to increase as the viscosity decreased. These uncommon features can now be justified by new mathematics. First, instability analysis (Gallay/Smets Anal. PDE Vol. 13, 2020) shows that Gaussian profiles are unstable by applying the traditional formalism for boundary layers to the shear around vortex cores. The example algebraic profile is largely not unstable. The domain was increased due to mathematics that shows that for a fixed configuration, at very small critical viscosity the energy dissipation rate goes to zero, a restriction empirically broken by using larger domains. The new mathematics shows that extensions of the original proof do not prevent this, and might require it.   

Time Resolved Particle Image Velocimetry Measurements of Non-Isothermal Vortex Breakdown Onset

Vortex breakdown occurs in many flow applications and controlling breakdown onset is often desirable. While there is an abundance of research into the behavior of vortex breakdown, there is limited understanding of the mechanisms that drive this process and the end state, especially in non isothermal flows. Previous research has motivated this work by suggesting that blue fire whirls could be controlled through heat addition at the breakdown bubble. The current work aims to experimentally show the effects of heating on the onset and structure of vortex breakdown in an incompressible non-reacting vortex flow. A new vortex breakdown rig has been developed to allow for high-speed particle image velocimetry and heating at the recirculation bubble. Time-resolved flow field measurements of the breakdown flow structures were taken with varying levels of heat addition, focusing on the process and onset of breakdown. The effects of the added heat on the breakdown characteristics were evaluated and compared to computational results. The results shed light on the experimental viability of controlling vortex breakdown via heat addition.

    

Not Participating

We perform resolvent analysis over a streamwise slice of the far-wake flowfield downstream a finite-span wing of an aspect-ratio of 2.5 and at 5 degree angle of attack.  At the chosen Reynolds number of 1000 and the free-stream Mach number of 0.3, the flow exhibits no unsteadiness.  The flowfield over the streamwise slice, characterized by a pair of counter-rotating tip vortices and a wake deficit in the streamwise velocity component, is considered as the base flow in the resolvent analysis. The analysis is performed in a parallel-flow setting where the streamwise wavenumber is parametrized.  At a low-frequency range, we observe that the dominant response modes are associated with the long-wave Crow instability.  Over this range, both the forcing and response modes show symmetric structures in the streamwise and transverse velocity components and anti-symmetric in the spanwise component.  According to the forcing mode structures, the Crow instability can be effectively triggered by introducing streamwise perturbation at the mid-span (symmetry) plane.  Over the high-frequency range, we do not observed the manifestion of Widnall instability.  This can be attributed to the low ellipticity in the shapes of the tip vortices.  Comparing these results to those from Lamb-Oseen vortex pairs, we also suspect the absence of Widnall structures from the dominant modes are likely due to the stabilizing effects of the streamwise velocity component in the wake.

    

Viscous Perturbation to Inviscid Wake Vortices - Perturbation Theory in Vortex Stability

We investigate how viscosity affects the linear stability of a q-vortex, the typical model of aircraft wake vortices, by means of the degenerate and non-degenerate perturbation theories that were originally developed for quantum mechanics for studying how energy levels of atoms and molecules changed subject to perturbations. For the regular, inviscid eigenmodes of the q-vortex, we verify that the non-degenerate perturbation theory yields the same viscous eigenmodes and eigenvalues that can be found numerically without the use of perturbation theory by solving for the eigenmodes of the full, viscous Navier-Stokes equations. For the singular, inviscid eigenmodes with singularities at their critical layers, it was found in our numerical analysis that the corresponding viscous eigenmodes appeared in two bifurcated sets of eigenmodes with slightly different complex growth rates. We quantitatively describe the reason for the bifurcation of the viscous eigenmodes using the degenerate perturbation theory and relate it to the two-fold degeneracies in the inviscid critical-layer eigenmodes. This approach, however, lacks the regular information inside the critical layers as it treats them as singular points. As a complement, asymptotic expansion of the critical layers should follow.   

Formation and Stability of a Ground Vortex in the Cross Flow over an Axisymmetric Inlet

The formation mechanisms of a ground vortex by cross flow along a plane surface and suction into an inlet of a cylindrical nacelle whose axis is normal to the wind direction and parallel to the ground plane is explored in wind tunnel experiments. The evolution of the columnar vortex and its ingestion into the inlet are investigated over a broad range of the primary formation parameters that include the speeds of the inlet and cross flow and the cylinder's elevation above the ground plane. The interactions between the inlet and cross flow and the formation of the ground vortex are measured using planar PIV in multiple planes normal to the nacelle's axis and parallel to the ground plane. The present investigations show that the vortex forms when the ratio of the inlet and cross flow velocities exceeds a critical value that varies linearly with the ratio of the elevation of the inlet's centerline and its diameter, as already pointed out in the past. It is also shown that when the critical value of the inlet/cross flow velocity ratio is exceeded for a given elevation ratio, the ground plane first begins to intersect the stream surface of the flow into the inlet, thereby enabling surface vorticity to roll into a "lambda" structure of two counter-rotating longitudinal vortices that begin to be advected towards the nacelle's inlet. Depending on the inlet/cross flow velocity ratio, one of the unstable longitudinal vortices is suppressed and the dominant vortex forms the ground vortex that is ingested into the inlet. Furthermore, it is shown that for given velocity ratio and ground plane elevation the ensuing flow field is self-similar.   

Bursting vortex ring

We establish the relationship between the twisting number and the local twist rate of vortex lines on different vortex surfaces, which extends the helicity topological decomposition to closed vortex tubes with differential twist. By developing numerical construction and measurement methods for differential twist, and using the direct numerical simulation and vortex-surface field, we quantitatively study the vortex bursting caused by differential twist in vortex rings. Two twist waves with opposite helical chirality collide on the cross section of the vortex ring after moving towards each other. In the subsequent vortex bursting, local vortex surfaces are squeezed to form a disk-like structure coiled by strongly twisted vortex lines. We model the dynamics of vortex bursting using a Burgers-like equation to estimate the bursting time. At large Reynolds numbers or initial twisting, vortex reconnection occurs in the disk-like structures, generating smaller dipole vortex tubes. The reconnection significantly weakens the twisting of vortex lines near the disk to inhibit the secondary bursting.