Session L39: Vortex Dynamics and Vortex Flows: General

Reversed steady streaming generated by a free-moving magnet.

Steady streaming is produced using a small magnet that oscillates due to electromagnetic forcing. The magnet is kept afloat on a shallow layer of weak conducting electrolyte. The oscillatory motion is promoted by injecting an alternating electric current to the electrolyte, the current interacts with the magnetic field of the magnet generating a Lorentz force in the fluid whose motion drives the magnet. In our case study, the well-known vortical structures of the streaming rotate in opposite direction to previous works on the subject. This reversed streaming is attributed to a non-previously considered degree of freedom, namely, the coupling between the fluid and the free-moving body. Analytical one-dimensional and two-dimensional solutions give a simple insight into the observed dynamics of the oscillating magnet and the promoted reversed streaming, respectively. A quasi-two-dimensional numerical simulation reproduces the experimental observations satisfactorily.   

Toward a Hybrid Vortex Simulation Approach with Mode Decomposition

Turbulence contains vortices with a broad range of spatial and temporal scales, which leads to expensive simulations when resolving all features in these flows. Accordingly, data-driven methods that model system dynamics from existing data have recently been of great interest to researchers. This study proposes a hybrid vortex simulation approach that combines classic vortex methods with proper orthogonal decomposition (POD) or Koopman mode decomposition (KMD). We seek a data-driven model that balances efficiency and accuracy. To automate the selection of the model parameter values, we record total circulation, enstrophy, and angular impulse and solve their theoretical evolution equations. We first test the robustness of selecting each of the above three flow quantities to assess the accuracy of the approach. We then verify that POD/KMD eigenvalues recover the correct time scales of the flow systems. Finally, we analyze the model performance by simulating the merger of uniform, patch-like co-rotating vortex pairs under different Reynolds numbers. This approach has the potential to be beneficial for solving flows with fine features in large domains.   

Vorticity dynamics and Josephson-Anderson relation for flow over spheroid

The connection between drag and vorticity fluxes for flow over spheroid is investigated numerically using the Josephson-Anderson (JA) relation. The JA relation decomposes the instantaneous work done by the drag force into vorticity fluxes across potential streamlines. The decomposition is first verified in canonical conditions including laminar and turbulent flows over a sphere, and subsequently applied to the prolate spheroid. The contributions to the drag force from vorticity fluxes in different flow regions are evaluated and related to local flow structures. These contributions are encapsulated in the instantaneous and mean Huggins vorticity-flux tensors which appear, respectively, in the instantaneous and mean vorticity transport equations. Analysis of the instantaneous tensor underscores the role of vortex-induced separation for the three-dimensional, separated boundary layer around the spheroid. In the downstream wake, the transverse mean-vorticity transport, which is dominated by the turbulent flux, balances the streamwise total pressure gradient. The results provide a novel perspective for drag and three-dimensional separation on bodies of revolution from the viewpoint of vorticity dynamics.    

Instantaneous drag for flow through general curvilinear channels in terms of vortex dynamics: the Josephson-Anderson relation

The detailed Josephson-Anderson (JA) relation for classical fluids [1] equates instantaneous work by pressure drop over any streamwise segment of a general channel to wall-normal flux of spanwise vorticity, spatially integrated over that section. The potential flow with the same mass flux, as in Kelvin’s minimum energy theorem, appears as background field and incorporates information about channel geometry. We have generalized this result to streamwise periodic channels for convenient use in numerical simulations. Whereas the usual Neumann b.c. [1] create an unphysical vortex sheet in a periodic channel, we exploit instead Dirichlet b.c. to define the background potential. We show that the minimum energy theorem still holds and the JA relation again equates work by pressure drop to integrated flux of spanwise vorticity. The result holds for both Newtonian and non-Newtonian fluids and for general curvilinear walls.  We present some numerical results for our new formula. The relation arose in quantum superfluid theory and it holds also for external flows around solid bodies [2], related to works of Burgers, Lighthill, etc. Drag and dissipation are thus related very generally to vorticity structure and dynamics locally in space and time, with many applications to drag-reduction, e.g. by polymers, and calculation of drag, e.g. in rough-wall channels.

[1] E. R. Huggins, Phys. Rev. A 1, 332-337 (1970) 

[2] G. L. Eyink, Phys. Rev. X 11, 031054 (2021)    

Vortical flow characteristics derived from local flow geometry and relationships to bundle of vorticity lines in organization of vortical structure

The present study investigates geometrical features of vortical flow structure and relationships to geometry of bundle of vorticity lines passing through a vortical region. A Galilei invariant coordinate system, i.e., vortex space associated with the local flow geometry, is defined in a vortex center, where the swirl plane and directions of both elliptic vortical flow and inflow/outflow are specified. The local flow geometry at the center derives primary geometrical features of radial and azimuthal flows as specific quadratic forms around the center in the swirl plane of a finite scale vortex, and the vortex space derives universal expression of the velocity gradient tensor where the gradient of an axial flow in the plane is given. If a bundle of vorticity lines swirls in the plane, this geometry operates the second derivative of the axial flow, and induces it even if the pressure gradient gives force in the opposite direction. While the vortical flow structure generates the geometrical features of the bundle of vorticity lines in a vortical region, the bundle contributes to the generation of the axial flow. A vortical analysis in homogeneous isotropic turbulence with Direct Numerical Simulation shows these complementary characteristics.   

Vorticity Layer–Vortex Interaction, a Prelude to Understanding Entrainment

Toward understanding the entrainment mechanism across the turbulent non-turbulent interface, we explore numerically the interaction between a vortex rod adjacent and aligned with a vorticity layer (VL). An Oseen vortex (Re_Γ=1000) is positioned near the VL with a circulation per unit length of  Re_s=1000. The VL thickness and the distance between the vortex center and VL are 1 and 2 times the vortex radius R, respectively. The increase in the vortical fluid volume is used to quantify the entrainment rate. When the vortex is orthogonal to the VL vorticity, the VL is distorted by the vortex, but the latter remains unperturbed. Additionally, in this configuration, large gradients of enstrophy enhance viscous diffusion and amplify the volume-integrated enstrophy 6 times the initial value. When VL and vortex vorticity vectors are parallel – hence no stretching or tilting – the entrainment rate is 11% less than when the vectors are orthogonal. Although vortex stretching dominates the interaction, there is, however, an initial transient entrainment rate reduction due to vorticity compression. Subsequently, a linear growth rate (1.8Γ/R²) of the enstrophy volume integral is acquired, dominated only by vortex stretching. Variation of enstrophy production both along and across VL is explained in terms of the induced velocity of the vortex.   

Generation of large-scale swirling flows and vortex breakdown in multifan wind facilities

With the recent growing interest in Advanced Air Mobility (AAM), the need to test and validate the flyability of these vehicles in vortical winds has become imperative for manufacturers and regulatory agencies (FAA). Multifan wind facilities (windshapers) are now commonly used to test free-flying vehicles in repeatable and controllable wind patterns. In particular, large-scale swirling flows can be generated using multiple windshapers set at an angle with respect to each other. The swirling flow morphology can be finely tuned by adjusting the azimuthal and axial flow conditions. For suitable flow regimes, vortex breakdown has been observed. The vortex bursting phenomenon can be explored with high precision and controllability thanks to the degrees of freedom in regulating the various flow components. In the present experimental setup, the onset of vortex breakdown occurs at a fixed critical value for the ratio of the rotational to the axial volume fluxes. The phenomenon exhibits no hysteresis.   

Finding Beauty in Asymmetry: Vortex Ring-Hemicylindrical Cavity Interactions

A three-dimensional vortex-ring impinging on a non-axisymmetric cavity produces asymmetric flow behavior. This interaction finds various engineering and biomedical applications. While symmetric vortex ring-cavity interactions have been shown to depend on cavity size, the influence of cavity size on more complex asymmetric interactions is not well understood. Elucidating these physics will enrich fundamental knowledge of vortex ring dynamics while providing insight into specific scenarios of interest; namely, replacement voice via tracheoesophageal speech. The goal of this study is to investigate the effect of cavity size on the physics of a vortex ring impinging on a hemicylindrical cavity. Vortex rings with formation number   and Reynolds number  were generated in a water tank. Six different ratios of vortex ring radius R_v to hemicylindrical cavity radius  were examined; namely, γ  = 1⁄4, 1⁄3, 2⁄5, 1⁄2, 2⁄3, & 1. Flow visualization and particle image velocimetry were performed. Due to the asymmetric impact, significantly different flow physics were observed in the two primary planes of interaction (i.e., the axial and radial planes). In the axial plane, the induced secondary vortex ring moved away from the impact surface instead of rotating around the primary ring, and the trajectory of the secondary vortex ring varied as a function of γ. In contrast, in the radial plane the secondary vortex ring completely rotated around the primary ring and moves towards the bottom of the hemicylinder surface throughout the interaction.   

Vortex ring generation by a piston with non-uniform velocity

Vortex rings are involved in many industrial and natural flows. The ease with which they can be generated in laboratory make them a much-studied object of research. The most conventional technique involves sliding a piston inside a cylinder to expel a volume of fluid which rolls up at the cylinder exit in the form of a vortex ring that detaches and propagates by self-induction. Although this method is commonly used, questions remain concerning the dynamics and final characteristics of the generated vortex ring, particularly in the case of a piston moving with a non-uniform velocity. In this context, the aim of the present study is to analyze the influence of the piston velocity law on the vortex ring characteristics, and to identify the physical mechanisms involved. Direct numerical simulations of vortex ring generation at the outlet of an orifice are performed with OpenFOAM. The motion of the piston is explicitly simulated using a dynamic mesh technique. A parametric study is carried out in the case of a velocity law consisting of a succession of two constant velocities separated by an ascending or descending step. The results show that the sign and amplitude of the step control the dynamics of the generated vortex rings. It is found that their propagation speed scales with the space-averaged velocity of the piston, and that their circulation and velocity are optimal for a particular step amplitude.   

Internal reflection of a vortex ring at an air-water interface

A vortex ring is a structure of fundamental importance in fluids, which can also be observed in other fields, including plasmas and electromagnetism. The interactions of a vortex ring with a solid boundary, a free surface, and other vortex rings have been widely studied. However, the intriguing phenomenon of reflection of a water vortex ring at an air-water interface has not drawn enough attention. We investigate this problem experimentally and observe vortex ring reflection that is similar to the total internal reflection of light. Based on our observations, we develop a simple model that explains the phenomenon.   

Stability of co-axial vortex rings with implications for circumstellar clumping

The Crow instability stimulates the growth of perturbations along two interacting vortex cores. A symmetric perturbation of the vortex core position grows until its amplitude is on the order of the core separation distance, at which point the cores touch and trigger a complex vortex reconnection process that results in isolated secondary vortex structures. While generally considered in the context of wingtip vortices, other interacting vortex cores, such as those induced by the Kelvin-Helmholtz instability in circumstellar disks, may also be Crow unstable. The ensuing perturbation growth would stimulate the formation of clumps of circumstellar material, possibly promoting the formation of planetesimals. In this study, we analyze the stability of two co-axial vortex rings of equal strength. Our theory predicts dominant unstable wavenumbers, expected to set the number of clumps along the cores, consistent with the observed number of clumps along existing circumstellar disks.   

The Big Effects of Small Vertical Velocities on the Structures of 3D Rapidly-Rotating, Stratified Planetary Vortices

Vortices in rapidly-rotating, stratified flows are often computed with 2D quasigeostrophic or shallow-water equations. However when the Rossby number is small and/or N/ω_z is large, the vertical velocity v_z of the vortex can strongly affect vertical vorticity ω_z of the vortex and the log of the temperature field of the vortex because their temporal derivatives are proportional to f v_z/Λ  and v_z N/g, respectively, where N is the Brunt-Vaisala frequency, g the acceleration of gravity, f the Coriolis parameter, and Λ the vertical density scale height. Here, we present some of the consequences of v_z for anticyclones computed with a 3D, high-resolution spectral code used to simulate the Jovian vortices. We show how v_z produces the bright warm rings observed around many of Jupiter’s anticyclones, such as the Great Red Spot, and how v_z increases the longevities of the large Jovian anticyclones to tens or hundreds of years -- far greater than the vortex turn-around times and the local atmosphere’s radiative decay time. We present a new analytic scaling relation for the magnitude of v_z in terms of the vortex's Rossby number and its vertical aspect ratio (i.e., its vertical scale height divided by the horizontal scale height of the pressure anomaly of the vortex) for vortices in which the vertical scale-heights of the vertical and horizontal components are of the same order.