Session A38: Vortex Dynamics and Vortex Flows: Fundamentals

Generating periodic vortex pairs using flexible structures.

When a planar flow starts through a narrow slit, it creates a unique mass of fluid characterized by counter-rotating vortex pairs. These vortex pairs typically remain attached to the fluid layer at the tip without separating in a phenomenon known as "pinch-off," usually observed in 3D vortex ring formation process. Our research aims to trigger this pinch-off in vortex pairs using flexible slit plates, enhancing momentum transport and self-propulsion. By employing a flow evolution model, we demonstrate that the growth rate of these ejected vortex pairs is proportional to the square root of time. We discovered a critical case of plate flexibility which induces pinch-off in the vortex pairs — a phenomenon not observed with rigid plates (slit). This setup generates a continuous sequence of vortex pairs, with their generation period closely matching the oscillation period of the plates. The streamwise velocity of the leading vortex pair varies non-monotonically with Ca, increasing up to Ca ≈ 0.04 before decreasing due to the upstream movement of smaller vortices. These findings provide new insights into the control and induction of vortex pair behavior, opening up innovative possibilities for fluid transport and advanced flow manipulation techniques.   

Vorticity dynamics in separated bluff-body flows

The back-in-time evolution of vorticity in flows over bluff bodies is studied, with particular focus on separation and reattachment. The precursors of vorticity at these locations are decomposed into interior and the boundary contributions. The boundary term was credited by Lighthill for the generation and removal of vorticity, in favorable and adverse pressure gradients, respectively. Separation in two-dimensional flows is then viewed as the result of sustained adverse pressure gradient. An element is missing from this description, which we expose and quantify in our analysis. Furthermore, Lighthill's interpretation does not generalize to three-dimensional separation, for example for the vorticity dynamics near separation on a prolate spheroid at incidence. Our analysis shows that significant cancellations take place between different components of the interior vorticity prior to reaching the separation and reattachment lines. In addition, vorticity tilting enables the transport of vorticity across the separation surface—an effect that is not possible in the two-dimensional case.   

Vortex formation length in the two-dimensional wake of a circular cylinder

Vortex formation length (VFL) has been used over the past 75 years as a characteristic length scale for defining the spatial extent of initial wake development behind a bluff body. Early considerations of VFL were informed by estimates of the vorticity field, but in practice the historical definitions are based on features of the time-averaged velocity or pressure fields, with each giving a single characteristic length at a fixed value of Reynolds number. We investigate the use of Proper Orthogonal Decomposition (POD) to extract information about vortex formation from the time-dependent flow, with a focus here on low Reynolds numbers (50<Re<200) so that the flow can be reasonably modeled as 2D. We compare results based on the velocity and pressure fields with those from the vorticity field as quantified by the lambda-2 criterion. We use the leading modes from the POD analysis to define upper and lower bounds on the VFL as a function of Reynolds number. Identifying a range, rather than a single value, for the VFL at a particular value of Reynolds number is consistent, both physically and quantitatively, with the hysteresis observed in the critical spacing of two in-line tandem cylinders.   

Forces on a body with a vortex-dominated wake

When a solid body is immersed in fluid moving at a constant speed at Reynolds numbers greater than about 50, a series of alternating vortices forms in its wake, known as the 'von Karman street'. The drag or thrust force on such a body can be predicted using von Karman's drag law. This law relates the characteristics of the vortex street in the wake to the drag (or thrust) coefficient on the body, and was derived by von Karman in 1911 using Newton's second law applied to a rectangular control volume enclosing the body and its wake. In this talk, we will present a generalization of von Karman's drag law to the case of an N-vortex street wake, i.e., to vortex-dominated wakes in which there are an arbitrary number of vortices in the wake. We will show that under a small set of assumptions, it is possible to relate the drag coefficient on a body with an N-vortex wake to the characteristics of the vortex street, including the strengths and positions of the vortices. We will apply this method to wakes with 3 or 4 vortices per period, corresponding to the experimentally observed 'P+S' wakes and 2P wakes.   

The N-vortex Problem in Doubly-periodic Domains with Background Vorticity

We study the N-vortex problem in a doubly periodic rectangular domain in the presence of a background vorticity field. We first consider a constant background field and derive an explicit formula for the hydrodynamic Green's function using a conformal mapping approach. We show that the point vortices form a Hamiltonian system and that the two-vortex problem is integrable. Several fixed lattice configurations are obtained for general N, some of which consist of vortices with inhomogeneous strengths and lattice defects. We then consider a smooth background field given by the Liouville PDE and show example solutions in which point vortices exist in stationary equilibrium with the background.   

Re-visiting the Taylor-Green vortex up to and after t ∼ 4.5

The evolution of the classic Taylor-Green vortex, under both Euler and Navier-Stokes, is re-visited using recent numerical analysis developed for vortex reconnection. For ν ≡ 0 Euler doubly-exponential, non-singular growth of ∥ω(t)∥_∞ is shown, supported using nonlinear inequalities of vorticity moments, all of which point away from singular growth. For Navier-Stokes, for a short period around t ∼ 4.5 when the interior vortices cross, the higher-order vorticity moments Ω_m can be scaled using the viscosity as ν^1/4 in a process that sheds negative helicity vorticity sheets. Including finite- time ν-independent convergence of √νZ(t)=(V^1/2ν^1/4Ω₁)²  and ν^1/4∥ω(t)∥_∞ during the phase leading to reconnection. The origin of the ν^1/4 scaling comes from how the sheets expand to compensate for the ν^1/2 scaled compression around those crossings. However, the type of post-reconnection acceleration of enstrophy growth that could lead to a viscosity-independent energy dissipation rate does not occur due to the restrictions imposed by the (2π)³ periodic domain.    

Sensitivity analysis and optimization of initial conditions in two-dimensional unsteady vortical flows

Vortical flow interactions and instabilities can be harnessed for practical applications, for instance reducing the energy in complex trailing wakes. In many scenarios, the high dimensionality and non-linearity of the flow poses challenges for predicting the sensitivity of late-time flow metrics to key geometric parameters governing the initial flow state. Here we investigate these challenges and possible solutions in the context of 2D vortical flows. First, we implement both a tangent and adjoint methodology in a Julia code. The computation of the derivative terms in the vorticity-velocity Navier-Stokes equations is efficiently handled using automatic differentiation (AD) functions from Julia packages. Using the sensitivities obtained from these methods, we investigate the feasibility of performing gradient-based optimization of the initial vortical structures, in order to minimize key global cost functions evaluated within a later time window. Additionally, we explore the application of a reduced-order model (ROM) approach to address cases of unsteady vortical flows with chaotic behaviors where the traditional tangent or adjoint methods fail. The ROM models the dynamic behavior of sensitivities in the phase space, which helps determine the modal functions causing instability in the sensitivity functions. We will outline the design of this ROM and show preliminary results on its application to highly nonlinear flow problems, compared with traditional sensitivity techniques.   

Beyond Thin Lines: the Vortex Bundle Model

It is common for flows, both natural and artificial, to contain concentrated regions of vorticity. These "vortex lines" are observed in a range of systems and at various scales; from a stirred cup of coffee to hurricanes. As a result, there is a long history of using models with infinitesimally thin vortex lines to study the motion of fluid. However, these models neglect core details and thus some phenomena (like vortex stretching) cannot be explored. In this talk, I will present a "vortex bundle" model, which incorporates local core details and interaction with boundary flows. This model allows us to compute a smooth flow field everywhere in space while ensuring the flow is divergence free. I will also present some preliminary results of vortex evolution using this model.   

Kelvin wave exchange between toroidal vortices

We study the interaction of a circular vortex ring with a helico-toroidal vortex 

in an incompressible inviscid fluid. The vortices are thin tubes of equal 

cross-section and circulation. The helico-toroidal vortex lies on the surface 

of an immaterial torus (radius: r0, cross sectional radius: r1) and is 

characterised by the number of turns, q, that it makes round the torus 

centerline while making one turn round the torus symmetry axis to close 

on itself. The vortex ring and the immaterial torus have the same 

centerline. Under these initial conditions, the flow evolution depends only 

on q, r1/r0 and a (the cross-sectional radius of the vortices). Numerical 

simulations using the Rosenhead-Moore approximation to the Biot-Savart 

law show that for small r1/r0  (<0.2) the vortices translate along and rotate

around the torus symmetry axis while changing shape almost periodically: 

at regular intervals the helico-toroidal vortex becomes an almost circular 

vortex ring and vice versa. The linear speed of the vortices decreases and 

their angular speed increases with growing values of the parameters q and 

r1/r0. The capacity of the vortices to carry fluid decreases while their 

capacity to stir their surroundings increases as r1/r0 increases.   

Dynamics of vortex-ring/wall collisions with background rotation

We present direct numerical simulations of toroidal vortex rings interacting with a flat boundary in presence of a background rotation. It is known that, depending on the Reynolds number (the nondimensional vortex-ring strength) the separation of wall vorticity can produce secondary and tertiary rings which amplify the initial azimuthal instabilities.

In presence of background rotation, this dynamics are altered owing to the balance of angular momentum which induces faster (slower) toroidal flow within the ring cores when they shrink (expand) radially.

We study the dynamics of the collision for different rotation rates and explore all regimes, from weak rotation up to the one rotation-dominated in which the collision with the wall is inhibited and the initial impulse of the ring produces inertial waves.

The simulations are performed by integrating the Navier-Stokes equations, written in cylindrical coordinates using staggered, central, second-order accurate finite-difference discretizations.

The conde is run on the latest Nvidia GPU architectures which allow for high-resolution simulations (order 1 billion nodes) within a wall clock time less than 24 hours  using a single node.