Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves

Abstract

We describe flow-visualization experiments and theory on the two-dimensional unsteady flow of an incompressible fluid in a channel with a time-dependent indentation in one wall. There is steady Poiseuille flow far upstream, and the indentation moves in and out sinusoidally, its retracted position being flush with the wall. The governing parameters are Reynolds number Re, Strouhal number (frequency parameter) St and amplitude parameter ε (the maximum fraction of the channel width occupied by the indentation); most of the experiments were performed with ε ≈ 0.4. For St ≤ 0.005 the flow is quasi-steady throughout the observed range of Re (360 < Re < 1260). For St > 0.005 a propagating train of waves appears, during every cycle, in the core flow downstream of the indentation, and closed eddies form in the separated flow regions on the walls beneath their crests and above their troughs. Later in the cycle, a second, corotating eddy develops upstream of the first in the same separated-flow region (‘eddy doubling’), and, later still, three-dimensional disturbances appear, before being swept away downstream to leave undisturbed parallel flow at the end of the cycle. The longitudinal positions of the wave crests and troughs and of the vortex cores are measured as functions of time for many values of the parameters; they vary with St but not with Re. Our inviscid, long-wavelength, small-amplitude theory predicts the formation of a wavetrain during each cycle, in which the displacement of a core-flow streamline satisfies the linearized Kortewegde Vries equation downstream of the indentation. The waves owe their existence to the non-zero vorticity gradient in the oncoming flow. Eddy formation and doubling are not described by the theory. The predicted positions of the wave crests and troughs agree well with experiment for the larger values of St used (up to 0.077), but less well for small values. Analysis of the viscous boundary layers indicates that the inviscid theory is self-consistent for sufficiently small time, the time of validity increasing as St increases (for fixed ε).