Stefan Llewellyn Smith
9500 Gilman Drive, La Jolla CA 920930411, USA
Department of Mechanical and Aerospace Engineering, University of California
Publications:
Chu T., Llewellyn Smith S. G.
Helical Contour Dynamics
2021, vol. 26, no. 6, pp. 600617
Abstract
The equations of motion for an incompressible flow with helical symmetry (invariance
under combined axial translation and rotation) can be expressed as nonlinear evolution
laws for two scalars: vorticity and alonghelix velocity. A metric term related to the pitch of
the helix enters these equations, which reduce to twodimensional and axisymmetric dynamics
in appropriate limits. We take the vorticity and alonghelix velocity component to be piecewise
constant. In addition to this vortex patch, a vortex sheet develops when the alonghelix velocity
is nonzero.We obtain a contour dynamics formulation of the full nonlinear equations of motion,
in which the motion of the boundary is computed in a Lagrangian fashion and the velocity field
can be expressed as contour integrals, reducing the dimensionality of the computation. We
investigate the stability properties of a circular vortex patch along the axis of the helix in the
presence of a vortex sheet and alonghelix velocity. A linear stability calculation shows that
the system is stable when the initial vortex sheet is zero, but can be stable or unstable in the
presence of a vortex sheet. Using contour dynamics, we examine the nonlinear evolution of the
system, and show that nonlinear effects become important in unstable cases.

Llewellyn Smith S. G., Chang C., Chu T., Blyth M., Hattori Y., Salman H.
Generalized Contour Dynamics: A Review
2018, vol. 23, no. 5, pp. 507518
Abstract
Contour dynamics is a computational technique to solve for the motion of vortices
in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours
is followed, and the velocity field moving the contours can be computed as integrals along
the contours. Its bestknown examples are in two dimensions, for which the vorticity between
contours is taken to be constant and the vortices are vortex patches, and in axisymmetric
flow for which the vorticity varies linearly with distance from the axis of symmetry. This review
discusses generalizations that incorporate additional physics, in particular, buoyancy effects and
magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics
structure. The extra physics can lead to timedependent vortex sheets on the boundaries, whose
evolution must be computed as part of the problem. The nonBoussinesq case, in which density
differences can be important, leads to a coupled system for the evolution of both mean interfacial
velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities
are materially conserved and whose evolution governs the flow.

Llewellyn Smith S. G., Nagem R. J.
Vortex Pairs and Dipoles
2013, vol. 18, no. 12, pp. 194201
Abstract
Point vortices have been extensively studied in vortex dynamics. The generalization to higher singularities, starting with vortex dipoles, is not so well understood.We obtain a family of equations of motion for inviscid vortex dipoles and discuss limitations of the concept.We then investigate viscous vortex dipoles, using two different formulations to obtain their propagation velocity. We also derive an integrodifferential for the motion of a viscous vortex dipole parallel to a straight boundary.
