Theoretical Analysis of Oscillating and Streaming Fields Between Two Parallel Beams and Numerical Investigation of the Cooling Efficiency

Abstract

The main purpose of this paper is to investigate the oscillating and streaming flow fields and the heat transfer efficiency across a channel between two long parallel beams, one of which is stationary and the other oscillating in standing wave form. The oscillating amplitude is assumed much smaller than the channel height. When the Reynolds number, which is defined by the oscillating frequency and the standing wave number, is much greater than unity, boundary layer structures are found near both beams, which are separated by a core region in the center of the channel. The oscillating fields within the core region and both boundary layers are obtained analytically. Based on the oscillating fields, the streaming fields within both boundary layers are also analytically obtained. Further investigation of boundary layer streaming fields shows that the streaming velocities approach constant values at the edges of the boundary layers and provide slip velocities for the streaming field in the core region. The core region streaming velocity field is numerically obtained by solving the mass and momentum conservation equations in their stream function–vorticity form. The temperature field is also computed for two cases: both beams are kept at constant but different temperatures (case A) or the oscillating beam is kept at a constant temperature and the stationary beam is prescribed a constant heat flux (case B). Cases of different channel heights are computed and a critical height is found. When the channel height is smaller than the critical value, for each half standing wavelength distance along the beams, two symmetric eddies are observed, which occupy the whole channel. In this case, the Nusselt number increases with the increase of the channel height. After the critical value, two layers of asymmetric eddies are observed near the oscillating beam and the Nusselt number decreases and approaches unity with the increase of the gap size. The abrupt change of the streaming field and the Nusselt number as the channel height goes through its critical value may be due to the bifurcation caused by instability of the vortex structure in the fluid layer.