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

Abstract

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.