Low-reynolds-number Instability of the Laminar Flow between Wavy Walls

Instability of a viscous incompressible flow in a channel with wavy walls is investigated theoretically, numerically and experimentally. Linear stability analysis shows that appropriately chosen wall waviness leads to flow destabilization at surprisingly low Reynolds numbers. The unstable mode of disturbances forms a vortex array, which travels downstream. The remarkable feature is that the most destabilizing waviness does not introduce any additional flow resistance. The outcome of the stability analysis are consistent with the result of direct numerical simulation obtained using CFD finite volume package FLUENT (Ansys Inc.). Preliminary experimental data gained for a channel with appropriately corrugated wall seem to confirm these predictions. NOMENCLATURE Symbol 2H Description Unit Channel height m p Pressure Pa Re Reynolds number S Wall corrugation amplitude QV Volumetric flow rate m/s u, v, w Velocity components m/s Wmax Maximum velocity m/s V ′ Fluctuation of velocity m/s α Corrugation wave number μ Dynamic viscosity Pas INTRODUCTION Enhancement of mixing in the laminar regime is of fundamental importance in numerous applications in microfluidics, biotechnology, medicine and heat transfer [1]. Significant improvement of mass and/or heat transfer can be achieved only if sufficiently complex and time-dependent vortex structures are present. Such structures can be triggered by various geometric modifications (e.g., the wall waviness or surface-mounted obstacles), external forcing (e.g. oscillations of a driving pressure gradient) or the combination of both [2,3]. Unfortunately, in most cases the mixing improvement is accompanied by large increase of hydraulic resistance. The current work describes the mixing-enhancement method based on the idea of forced chaotic convection in the channel with appropriately shaped and transversely oriented wall waviness. The method is based on results of linear stability analysis applied to a simple unidirectional flow in a wavy channel. It was demonstrated in [4,5] that the laminar flow in the channel with properly tuned, transversely-oriented wall waviness can spontaneously loose stability at the Reynolds numbers as low as 60. Following this idea, the threedimensional numerical simulations of viscous flow through wavy channel are performed both for the infinite flow domain (periodic boundary conditions) and for the channel bounded by the side walls. The positive outcome of the numerical computations gave us confidence to the theoretical predictions, necessary to design microchannel mixer. Finally, the flow structures observed for laminar flow of water in a 33.6 mm wide wavy microchannel are investigated to identify predicted instabilities. STABILITY ANALYSIS Consider the reference case: the laminar incompressible flow in the region between two parallel planes Y = -H and Y = H. Let GP < 0 denotes the constant pressure gradient which drives the fluid in the positive direction of the Z axis. The velocity field can be expressed as