On the appearance of Moffatt eddies in viscous cavity flow as the aspect ratio varies

Abstract

Two-dimensional Stokes flow in a rectangular driven cavity is studied. Flow in very tall or shallow cavities has many counter-rotating eddies lying along the cavity centerline. This structure is investigated by constructing asymptotic approximations to the flow based on the assumptions A Ӷ 1 and A ӷ 1, where A is the cavity's aspect ratio. We show that the number of eddies increases as A tends to infinity or zero and derive asymptotic formulas for the values of the aspect ratio at which the streamline topology bifurcates and a new eddy appears. We concentrate particularly on flow driven by translation of the top and bottom cavity walls with equal and opposite velocities. For this benchmark problem our asymptotics are able to connect existing computations ͑performed in the approximate range of 10 −1 Յ A Յ 10 1 ͒ with Moffatt's 1964 theory for an infinite channel ͑i.e., A =0 or ϱ͒. We show that tall cavities have sequences of eddies which match the infinite channel flow, as observed in the previous computations. In contrast, shallow cavities have only half the number of eddies and a more complicated streamline topology. In both cases our asymptotic approximations give analytic formulas for the number of eddies and the shape of the streamlines. Other flows, corresponding to different driving at the boundaries, are also discussed and can be treated by the asymptotic methods we derive.

Physical modeling of vortical cross-step flow in the American paddlefish, Polyodon spathula

“…Our data on vortex parameters indicate that the vortices generated continuously downstream from backward-facing steps in the preserved paddlefish and the 3D models have the characteristics of forced (rotational) vortices rather than free (irrotational, potential) vortices [ 49 ]. Forced vortices are common at low to moderate Re (e.g., cavity flow and Moffatt eddies [ 50 – 52 ]; stable attached eddies generated by bluff bodies [ 53 ]). In a forced vortex, a continuous external source of kinetic energy must maintain the speed of the fluid throughout the vortex [ 54 ].…”

Microscopic velocity field measurements inside a regular porous medium adjacent to a low Reynolds number channel flow

“…17 characterised by viscous eddies and flow separation at sharp corners, even at creeping flow conditions. [36][37][38] The knowledge of the microscopic flow field in a porous medium and free-flow coupled system is however the key for gaining insight into the developing models, as well as the pore scale physical processes, such as convective transport. For example, Direct Numerical Simulations (DNS) showed that different topological features exist in a porous medium, 39 while their further influence on convective transport from the porous surfaces has been only numerically evaluated.…”

Microparticle image velocimetry (μPIV) study of microcavity flow at low Reynolds number

“…For all the studied geometry aspect ratios (λ = 0.25-3) in our experiments, there is only one single central vortex appeared in the microcavities, even at relatively high Reynolds number (Re = 180) when we increased the flow rate trying to observe corner vortices. On the contrary, corner vortices and additional vortices have been identified in previous reports related to lid-driven closed cavities on a macroscale (Heaton 2008;Shankar 1997;Shen and Floryan 1985;Chiang et al 1996). For example, two corner vortices occur at the bottom of cavity with aspect ratio λ = 4, while two central vortices appear inside a cavity with aspect ratio λ = 0.5 (Shen and Floryan 1985).…”

“…Historically, low Reynolds number flows in cavities, characterized by separating and recirculating flows have been extensively characterized (Heaton 2008;Moffatt 1964;O'Brien 1972;Shankar 1997;Shankar and Deshpande 2000;Shen and Floryan 1985). Most of the research work has been devoted to study of flow separation on macroscale, and only a few researches are focused on the flow behaviors in the microcavities (Yu et al 2005).…”