Influence of wall shape on vortex formation in modulated channel flow

Abstract

The flow inside channels with periodic, wavy walls of arbitrary shape is considered numerically. Solutions are obtained using either a perturbation approach, for weak modulation amplitude, or a finite volume technique, for strong amplitude. The flow is examined for sinusoidal, arched and triangular modulation over a wide range of amplitude, wavelength and Reynolds number in the steady laminar regime. For weak wall modulation (ε<0.3, α<2, where ε and α are the dimensionless half-wave height and wavelength, respectively), it is found that the flow behavior along the modulated wall is of the boundary-layer type. As such, the critical Reynolds number, Rec, for separation for each modulation shape can be expressed as an explicit function of ε and α, while the location of separation and pressure distribution along the modulated wall scale with ε, α, and Rec. For strong modulations, the boundary layer model is no longer satisfactory to predict the flow behavior and deviations from the trends found for weaker modulations are observed. It is also shown that the driving force required to sustain a given flow rate increases as ε increases. For all modulation amplitudes, the sinusoidal wave shape is found to require the largest pressure gradient to maintain a given flow rate through the channel and, consequently, yields the highest friction factor. Finally, the existence of a stable recirculating flow regime is discussed in the light of earlier stability analyses.

A note on the onset of recirculation in a 2D Couette flow over a wavy bottom

“…The flow is actually time-varying since the initial condition is inferred as the Couette flow given by Eq. (11), and the simulations stopped when a quasi-stationary state was reached. The criteria used to stop the simulations were based on the variation of the maximum velocity dv max throughout the domain between two time steps (usually reached before the scaled dimensionless time t * h/λ ≈ 120).…”

“…Nakayama and Sakio 10 studied numerically, through DNS, a pressure driven flow over a wavy bottom defined in terms of two modes of two-dimensional cosine waves, with different amplitudes and wavelengths at the lower boundary in order to explore the effects of filtering the small-scale fluctuations of the flow and the effects of smoothing of the boundary conditions on large eddy simulation (LES). Zhou et al 11 studied Poiseuille flow using a perturbation technique for small wave amplitudes and a finite element numerical code to investigate large perturbations for sinusoidal, triangular, and arched-shaped channels with a flat bottom. Sobey 12 has studied the Poiseuille flow using the triple deck theory in the asymptotic limit Re → ∞ using a first order approximation, obtaining that the critical (a/h) number for the onset of recirculation is (in our notation)…”

On the interplay between inertial and viscoelastic effects for the flow in weakly modulated channels

“…A correlation between shear stress proÿles near the walls and the appearance of back ow does not appear to exist for viscoelastic uids. It is worth mention that in addition to Zhou et al [13,14], Tsangaris and Leiter [32] demonstrated that for a Newtonian uid, the switch in the xy sign indicates the onset of back ow.…”

“…For Newtonian uid ow inside modulated channels, Zhou et al [13,14] compared the ÿrst-order perturbation results to simulations obtained using ÿnite volume approach [44]. The validity of the ÿnite-volume code itself was established by comparison with the experimental results of Nishamura et al [33,34].…”

Electroosmotically driven creeping flows in a wavy microchannel

“…At the same time, a vast amount of studies are found in classic fluid mechanics studies for the creeping motion in channels of varying geometries (Burns and Parkes 1967;Chow and Soda 1972;Wang 1978;Sobey 1980;Stephanoff et al 1980;Tsangaris and Leiter 1984;Pozrikidis 1987Pozrikidis , 1988Hemmat and Borhan 1995;Leneweit and Auerbach 1999;Wang 2003;Wei et al 2003;Zhou et al 2003;Scholle 2004;Luo and Pozrikidis 2006;Malevich et al 2006). Various analytical and numerical methods have been employed in these studies.…”