Session L16: Vortex Simulation and Theory

Interaction of two axially-symmetric helical vortices

We studied the evolution of two equal, coaxial, symmetrically located helical vortices in an unbounded, inviscid, incompressible fluid.  We computed the evolution with a triple-periodic vortex-in-cell method and found four main regimes of behavior: 1) Thin vortices translate and rotate uniformly while preserving their form, as predicted by Joukowsky (1912), and their velocities agree with those obtained by Velasco Fuentes (2018), 2) Thicker vortices behave as predicted by Joukowsky (1912) but they move with smaller velocities than the thin vortices studied by Velasco Fuentes (2018), 3) Fat, large-pitch vortices merge to form a single cylindrical column with mainly axial vorticity. This process is analogous to the merger of two Rankine vortices (which correspond to helical vortices of infinite pitch) but the vortices approach each other faster and merge even from relatively large distances (where Rankine vortices would not merge). 4) Small-pitch vortices merge to form a single cylindrical shell with mainly azimuthal vorticity. This occurs even for thin vortices if the pitch is sufficiently small.


  

A New Implementation of the Vortex Method for Elliptical Vortex Evolution

We discuss a new implementation of the vortex method for the incompressible Euler equations. This work focuses on smooth vorticity distributions as opposed to vortex sheets. As usual the vorticity is carried by Lagrangian particles and the velocity is recovered by the Biot-Savart law. The new implementation combines remeshing, adaptive refinement, and treecode acceleration to resolve complex features of the flow. Applications are given to vortex dynamics in two-dimensional free-space.    

A new type of axisymmetric vortex shedding from a sphere moving vertically in a stratified fluid

A numerical study is described of the flow past a sphere moving vertically at constant speed in a stratified fluid. It is found that the flow under moderate or strong stratification (Fr=W/Na ≤14, W: sphere velocity, N: buoyancy frequency and a: sphere radius) remains axisymmetric even at Re=500, unlike the flow of a homogeneous fluid which becomes asymmetric for Re>200. A striking feature is that this axisymmetric flow gives, under moderate stratification (7≤Fr≤14), an axisymmetric vortex or 'vorticity' shedding, which has never been observed in the wake of an axisymmetric bluff body. The 'vorticity' shedding occurs even when the familiar vortex shedding can not be identified in the streamlines.  The phenomenon corresponds to the observation in previous shadowgraph experiments (e.g. Hanazaki 2009), which showed a periodic generation of knots in the jet downstream of a sphere.
  

A non-Boussinesq vortex ring

The motion of an axisymmetric buoyant vortex ring is calculated using contour dynamics. An evolution equation for vortex sheet on the interface is obtained to account for additional physics such as buoyancy. In this talk, we will first review the Boussinesq limit. Then an equation for vortex sheet in the non-Boussinesq regime is derived. The evolution of rings using both approaches are compared for Atwood numbers. In the non-Boussinesq case, surface tension can also be introduced through a pressure jump on the interface. We also discuss the relevant dimensionless numbers and explore the dynamics of buoyant vortex rings in parameter space.   

Unsteady vortex interactions with Joukowski airfoils on elastic supports

Coherent vortices present in the atmosphere or that are generated by aircraft create a gust field that exerts unsteady aerodynamic forces on the wings and appendages of downstream aircraft. In particular, the unsteady aerodynamic forces resulting from spanwise-oriented gust encounters produce an unsteady vortex wake that is also coupled to the motion of the incident vortex and to the wing shape and position. These coupled interactions between an incident line vortex, a Joukowski airfoil on elastic supports, and its wake are formulated analytically and computed numerically. The vortex motion and the fluid forces on the airfoil are derived from inviscid potential flow, and the continuous shedding of vorticity from the trailing edge is modeled by the emended Brown and Michael equation. Special attention is paid to the limiting cases of flat-plate airfoils that are either stationary or under prescribed motions for which validation data exist. The present effort extends beyond these restrictions to include the geometrical effects of the Joukowski airfoil profile and its aeroelastic influence on the incoming vortex path and the unsteady fluid loading.   

The vortex-entrainment sheet

A vortex sheet is a well-known inviscid model of high Reynolds number viscous boundary and shear layers. The circulation in the layer is preserved via a jump in the harmonic potential across the sheet; there is also a jump in the tangential velocity that defines the vortex sheet strength. However, viscous layers necessarily contain mass and momentum, in addition to vorticity, which are entrained by a normal velocity at the edge of the layer. Hence, a more complete model of a viscous layer or wake that is collapsed to an infinitely thin sheet should account for the entrainment as well as the vorticity. We propose such a model, termed a vortex-entrainment sheet, which is characterized by jumps in the harmonic potential and the stream function or, equivalently, by jumps in both the tangential and normal components of velocity. For separation at a sharp edge, the singular pressure gradient is neutralized, in accordance with the normal momentum balance, by delivering an instantaneous impulse to the fluid. A non-tangential shedding angle of the sheet from the sharp edge necessarily requires non-zero entrainment.   

Helical contour dynamics

Contour dynamics simulates the evolution of vortices by following the evolution of their boundaries. In its two-dimensional version, the vorticity inside vortices is constant, while in its axisymmetric version, the vorticity is proportional to distance from the axis of symmetry. We investigate helical contour dynamics, for which the flow is invariant along a helical vector. The nonlinear inviscid equations of motion reduce to two advection equations with forcing on the right-hand sides. However, this forcing is only non-zero along the boundaries of vortices for appropriate chosen velocity distributions. This leads to time-dependent vortex sheets on the boundary. The resulting contour dynamics equations are derived and some example cases are studied.   

Hollow vortex in a corner straining flow

A hollow vortex is a solution to the 2D Euler equations with a finite-area region of constant pressure and nonzero circulation. Hollow vortices in straining flow have been previously treated, but the influence of boundaries has not yet been investigated. Equilibrium solutions for a hollow vortex in straining flow in a right-angled corner are found in terms of the Schottky-Klein prime function. Using complex-variable techniques and the prime function, the vortex boundary is constructed as a conformal map from the canonical doubly-connected domain of the concentric annulus to the physical domain of the hollow vortex in a corner. Comparison with the limiting case of a point vortex in a corner is made, and extensions to corners of arbitrary angle and non-equilibrium cases are discussed. Motivation includes regions of high nutrient or pollution density in the ocean, which can be formed in coastal regions from rivers or runoffs; the advection and mixing of the regions depend in part on the local flow and shoreline geometry, and a hollow vortex in straining flow is a possible model for this process.