Eddy genesis and manipulation in plane laminar shear flow

Eddy formation and presence in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one has a sinusoidal profile and is fixed. The governing equations are solved via a full finite element formulation for the general case and semi-analytically at the Stokes flow limit. The effects of varying geometry (involving changes in the mean plate separation or the amplitude and wavelength of the lower plate) and inertia are explored separately. For Stokes flow and varying geometry, excellent agreement between the two methods of solution is found. Of particular interest with regard to the flow structure is the importance of the clearance that exists between the upper plate and the tops of the corrugations forming the lower one. When the clearance is large, an eddy is only present at sufficiently large amplitudes or small wavelengths.

However, as the plate clearance is reduced, a critical value is found which triggers the formation of an eddy in an otherwise fully attached flow for any finite amplitude and arbitrarily large wavelength. This is a precursor to the primary eddy to be expected in the lid-driven cavity flow which is formed in the limit of zero clearance between the plates. The influence of the flow driving mechanism is assessed by comparison with corresponding solutions for the case of gravity-driven fluid films flowing over an undulating substrate. When inertia is present, the flow generally becomes asymmetrical. However, it is found that for large mean plate separations the flow local to the lower plate becomes effectively decoupled from the inertia dominated overlying flow if the wavelength of the lower plate is sufficiently small. In such cases the local flow retains its symmetry. A local Reynolds number based on the wavelength is shown to be useful in characterising these large-gap flows. As the mean plate separation is reduced, the form of the asymmetry caused by inertia changes, and becomes strongly dependent on the plate separation. For lower plate wavelengths which do not exhibit a cinematically induced secondary eddy, an inertially induced secondary eddy can be created if the mean plate separation is sufficiently small and the global Reynolds number sufficiently large.