In this work, we consider steady two-dimensional interfacial waves in a two-layer stratified fluid, which is induced by a vortex pair located in the lower layer of the fluids. A mathematical model based on the boundary integral equation method and the potential-flow theory is established. The linear analytical solution for the linearized model is given in the form of Cauchy integral and then asymptotic behaviour for large x is presented. The fully nonlinear model is solved by the Jacobian-free Newton–Krylov (JFNK) method numerically. Nonlinear characteristics of wave profiles are identified compared with the linear results under different vortex strengths and the distance between the vortex pair. The amplitude of steady downstream waves is found to vary periodically with respect to the distance of the vortex pair, which can be regarded as the interference between waves produced by each vortex. For equal-strength counter- and co-rotating pairs, the downstream wave heights of linear solutions can be eliminated for some special values of the distance between point vortices, namely, the destructive interference occurs. Meanwhile, the wave only exists between the vortex pair like trapped waves. So does the nonlinear counterpart for counter-rotating pairs, but it could not be diminished with any distance.

In this paper, we study the two-dimensional Navier-Stokes flows with time-periodic external forces of large strengths. Invariant solutions, including periodic orbits and relative periodic orbits, are extracted with the recurrent flow analysis, while low-dimensional projections based on the dynamic mode decomposition (DMD) algorithm are used to reduce the cost of searching nearly recurrences. When the period of forces gets a constant increase, the flows change from the stable time-periodic state to oscillate and even turbulent flows. In all cases, one periodic orbit is identified near the initial stage. This orbit represents the stable/unstable base state, and the trajectories of vorticity fields are trapped inside it or escape away from it, leading to oscillating/turbulent motions. For the oscillating flows, four calculated periodic orbits without any symmetries play the role that the flows visit them and then move away from them to other orbits. Besides, for a moderate period of forces, a bursting phenomenon occurs and the state of oscillating flows turns to turbulent flows with the rapid increase of energy. For the turbulent motions, one unstable periodic, which qualitatively represents the shapes of a large vortex dipole that exists in the turbulent motions, is obtained. Its statistical significance is shown by the frequency of that flows visit it.