Spiral waves in excitable media possess both wave-like and particle-like properties. When resonantly forced (forced at the spiral rotation frequency) spiral cores travel along straight trajectories, but may reflect from medium boundaries. Here, numerical simulations are used to study reflections from two types of boundaries. The first is a no-flux boundary which waves cannot cross, while the second is a step change in the medium excitability which waves do cross. Both small-core and large-core spirals are investigated. The predominant feature in all cases is that the reflected angle varies very little with incident angle for large ranges of incident angles. Comparisons are made to the theory of Biktashev and Holden. Large-core spirals exhibit other phenomena such as binding to boundaries. The dynamics of multiple reflections is briefly considered.

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.

We examine how known unstable equilibria of the Navier-Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number Pr, the ratio of viscosity to density diffusivity, and fixed moderate Reynolds number, Re = 400. In the two asymptotic limits Pr → 0 and Pr → ∞, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the Pr → 0 limit, density variations away from a constant stable density gradient become vanishingly small as diffusion of density dominates over advection, allowing equilibria to exist for bulk Richardson number Ri b O(Re −2 Pr −1 ). Alternatively, at high Prandtl numbers, density becomes homogenised in the interior by the dominant advection which creates strongly stable stratified boundary layers that recede into the wall as Pr → ∞. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing Ri b . An asymptotic analysis is presented in the passive scalar regime Ri b O(Re −2 ), which reveals O(Pr −1/3 )-thick stratified boundary layers with O(Pr −2/9 )wide eruptions, giving rise to density fingers of O(Pr −1/9 ) length and O(Pr −4/9 ) width that invade an otherwise homogeneous interior. Finally, increasing Re to 10 5 in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.

Resonantly forced spiral waves in excitable media drift in straight-line paths, their rotation centers behaving as pointlike objects moving along trajectories with a constant velocity. Interaction with medium boundaries alters this velocity and may often result in a reflection of the drift trajectory. Such reflections have diverse characteristics and are known to be highly nonspecular in general. In this context we apply the theory of response functions, which via numerically computable integrals, reduces the reaction-diffusion equations governing the whole excitable medium to the dynamics of just the rotation center and rotation phase of a spiral wave. Spiral reflection trajectories are computed by this method for both small- and large-core spiral waves in the Barkley model. Such calculations provide insight into the process of reflection as well as explanations for differences in trajectories across parameters, including the effects of incidence angle and forcing amplitude. Qualitative aspects of these results are preserved far beyond the asymptotic limit of weak boundary effects and slow resonant drift.