Viscous flows in the channels with sinusoidal wavy walls are considered. Fully developed two-dimensional steady flows and their linear stabilities are investigated numerically and experimentally. Thermal characterisitics of two-dimensional steady basic flows and the secondary flows resulted from the instabilities of the basic flows are analysed numerically.
Two-dimensional steady flows are numerically obtained by the streamfunction-vorticity formulation. As the Reynolds number increases, the flows are characterized by the appearance of flow separation and the growth of recirculation region.
Linear stability of two-dimensional steady flows for two different phase (φ = 0, π) is investigated through the direct numerical simulation of linearized stability equations. Three-dimensional disturbance equations are formulated by introducing the vector potential. It is found that, for the dimensionless periodic length 2L = 3.0, the critical Reynolds number is substantially lower than that for plane channel flow and that when non-dimensionalized wall variation amplitude $\epsilon$ is smaller than a critical value (about 0.26 for φ = π, 0.28 for φ = 0), critical modes are three-dimensional stationary and for larger $\epsilon$, two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of channel flows with φ = 0 are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point while the critical two-dimensional mode resembles closely the Tollmien-Schlichting wave for plane Poiseuille flow.
The transition phenomena in symmetric wavy channel are experimentally studied. Three different channel heights (2H = 12, 16, 19 mm) are considered. The two-dimensional steady state, as flow rate is increased, bifurcates to two-dimensional asymmetric unsteady flow state in case of the smallest channel height 2H = 12 mm, which corresponds to the largest $\epsilon$. It is observed that the wave length of unstable mode is a half of wavy wall periodic length (2L = 30 mm). For channels with the largest (2H = 19 mm) and middle (2H = 16 mm) height, on the other hand, the flows bifurcate to three-dimensional steady states. The spanwise periodic lengths of flow structures are about 1.6 L for largest height, 1.2 L for middle height channel. These trends of observed flow transitions are in accordance with the expectations from the linear stability analysis.
In the numerical analysis of thermal characteristics of wavy walled channel flows, the constant wall temperature condition is applied. The characteristics of two-dimensional steady flows and temperature fields such as friction coefficient and heat transfer rate from the walls are presented as a function of the Reynolds and Prandtl numbers, the amplitude of wall variations and the phase differences. In particular, the heat transfer performance of the wavy channel is compared with the flat plate channel under identical pumping power constraint. It is found that, as the wall variation amplitude, the Reynolds and Prandtl number is increased, the heat transfer rate ratio is monotonically augmented. The symmetric channel (φ = π) reveals a best performance as compared with non-symmetric wavy channels.
The thermal characteristics after flow instabilities are also studied numerically. The secondary flow fields are obtained by the superposition of basic states and linear modes, with the assumption that the amplitudes of linear modes are proportional to the square-root of $Re - Re_c$. We have confirmed, through the direct numerical analysis of two-dimensional nonlinear equations, that two-dimensional oscillatory instability causes a supercritical Hopf bifurcation which gives a square-root dependence of oscillation amplitude on the Reynolds number. The temperature fields under the secondary flows are obtained up to $O(\tau^2)$ using perturbation technique. At weekly supercritical states ($Re ≥ Re_c$), the Nusselt numbers always have higher values than those for two-dimensional steady flows with same Reynolds numbers and grow nearly linear pattern as $Re - Re_c$ increases. It is found that, under same τ conditions, the two-dimensional time-periodic flow fields cause larger heat transfer rates from the walls than those for three-dimensional steady flows.
