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Hydrodynamic effects in channel flow due to vibrating walls

The non-linear effects of transverse wall vibrations on a laminar channel flow is investigated. A Newtonian viscous fluid is confined in an infinite channel with flexible walls where the motion of the wall takes the form of either travelling or standing waves. Small amplitude vibrations are considered and a perturbation approach is utilized. The small parameter is the amplitude and the surface velocity for moderate and large Strohaul numbers respectively. A pseudo-spectral numerical method is employed to solve the resulting linear system of equations. Resonance is found in the oscillatory flow at a Strohaul number of approximately 0.6. This occurs due to interactions between the forced vibratory flow and a natural mode of the unperturbed flow. In the vicinity of the resonance the mean flow rate decreases strongly for waves travelling in the direction of the channel bulk flow and for standing waves. The impact of this phenomenon is, furthermore, enhanced with increasing Reynolds numbers. Increasing the Strohaul number above unity reveals both negative and positive effects on the total flow rate in the channel, though it is much weaker in comparison to the effect of resonance. A variety of distinct streaming velocity profiles are found depending on the parameters of the problem. For travelling waves these profiles always are independent of the downstream coordinate. The contrary holds for standing waves and for sufficiently large Strohaul numbers a cellular flow structure is always found at second order. Thus, at the right conditions a standing wave may produce separation of the flow. Different aspects of flow control is also discussed where surface motions can be applied.