The onset of three-dimensional standing and modulated travelling waves in a periodically driven cavity flow

Three-dimensional instabilities of the two-dimensional flow in a rectangular cavity driven by the simple harmonic oscillation of one wall are investigated. The cavity has an aspect ratio of 2:1 in cross-section and is infinite in the spanwise direction. The two-dimensional base flow has no component in the spanwise direction and is periodic in time. In addition, it has the same space-time symmetry as a two-dimensional periodically shedding bluff-body wake: invariance to a mid-plane reflection composed with a half-period evolution in time. As for the wake, there are two kinds of possible synchronous three-dimensional instability; one kind preserves this space-time symmetry and the other breaks it, replacing it with another space-time symmetry. One of these symmetry breaking modes has been observed experimentally. The present study is numerical, using both linear Floquet analysis techniques and fully nonlinear computations. A new synchronous mode is found, in addition to the experimentally observed mode. These two modes have very different spanwise wavelengths. In analogy to the three-dimensional instabilities of bluff-body wakes, the long-wavelength synchronous instability is named mode A, while that for the short wavelength is named mode B. However, their space-time symmetries are interchanged compared to those of the synchronous bluff-body wake modes. Another new, but non-synchronous, mode is found: this has complex-conjugate pair Floquet multipliers, and arises through a Neimark-Sacker bifurcation of the base flow. This mode, QP, has a spanwise wavelength intermediate between modes A and B, and manifests itself in the nonlinear regime as either quasi-periodic standing waves or modulated travelling waves.